INMEMORIAM 
FLORIAN  CAJORl 


i^l^ti 


^. 


^^' 


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1  -y 


AN 


ELEMENTARFrr;:- 


GE  OMETR Y 


AND 


TRIGONOMETRY 


WILLIAM    F.  BRADBURY,  A.  M., 

HOPKINS   MASTER  IN  THE   CAMBRIDGE  HIGH   SCHOOL;   AUTHOR  OF   A   TREATISE   ON  TMOOItOBTRrBT 
AND  SUaVETING,  AND  OF  AN  ELEMENTARY  ALGEBRA. 


BOSTON: 

PUBLISHED  BY  THOMPSON,  BROWN,  <fe  CO. 

23  Haavlky  Stkkkt. 


EATON    AND    BRADBURY'S 

gsBD.  wrnj' ^jatAMPLCD'.str'cfeEss  in  the  best  schools  and 
;/.  :  :/:  vj :  ;  AcJ^p^iipai^'pJ?'  the  country. 


Eaton's  Primary  Arithmetic. 
Eaton's  Elements  of  Arithmetic. 
Bradbury's  Eaton's  Practical  Arithmetic. 

Eaton's  Intellectual  Arithmetic. 
Eaton's  Common  School  Arithmetic. 
Eaton's  High  School  Arithmetic. 


Bradbury's  Elementary  Algebra. 

Bradbury's  Elementary  Geometry. 

Bradbury's.  Elementary  Trigonometry. 

Bradbury's  Geometry  and  Trigonometry,  in  one  volume. 

Bradbury's  Elesientary  Geometry.     University  Ediiim. 
Plane,  Solid,  and  Spherical. 

Bradbury's  Trigonometry  and  Surveying. 

Keys  op  Solutions  to  Practical,  Common  School,  and 
High  School  Arithmetics,  to  Elementary  Algebra, 
Geometry  and  Trigonometry,  and  Trigonometry  and 
Surveying,  for  the  use  of  2'eachers. 


COPYRICIIT,  1872. 

By  WILLIAM  F.   BRADBURY. 


CAJORI 


University  Press  :  John  Wilson  &  Son, 

CAMUUIDCe. 


PREFACE 


A  LARGE  number  of  the  Theorems  usually  presented  in  text- 
books of  Geometry  are  unimportant  in  themselves  and  in  no 
way  connected  with  the  subsequent  Propositions.  By  spending 
too  much  time  on  things  of  little  importance,  the  pupil  is  fre- 
quently unable  to  advance  to  those  of  the  highest  practical 
value.  In  this  work,  altliough  no  important  Theorem  has  been 
omitted,  not  one  has  been  introduced  that  is  not  necessary  to 
the  demonstration  of  the  last  Theorem  of  the  five  Books,  namely, 
that  in  relation  to  the  volume  of  a  sphere.  Thus  the  whole 
constitutes  a  single  Theorem,  without  an  unnecessary  link  in 
the  chain  of  reasoning. 

These  five  Books,  including  Ratio  and  Proportion,  are  pre- 
sented in  eighty-one  Propositions,  covering  only  seventy  pages. 
This  brevity  has  been  attained  by  omitting  all  unconnected 
propositions,  and  adopting  those  definitions  and  demonstrations 
that  lead  by  the  shortest  path  to  the  desired  end.  At  the  close 
of  each  Book  are  Practical  Questions,  serving  partly  as  a  review, 
partly  as  practical  applications  of  the  principles  of  the  Book, 
and  partly  as  suggestions  to  the  teacher.  As  those  who  have 
not  had  experience  in  discovering  methods  of  demonstration 
have  but  little  real  acquaintance  with  Geometry,  there  have 
been  added  to  each  Book,  for  those  who  have  the  time  and  the 
ability,  Theorems  for  original  demonstration.  These  Exercises, 
with  different  methods  of  proving  propositions  already  demon- 

'91139a 


IV  PREFACE. 

strated,  include  those  that  are  usually  inserted,  but  whose  dem- 
onstration in  this  work  has  been  omitted.  In  some  of  these 
Exercises  references  are  given  to  the  necessary  propositions ;  in 
some  suggestions  are  made ;  and  in  a  few  cases  the  figure  is 
constructed  as  the  proof  will  require. 

A  sixth  Book  of  Problems  of  Construction  is  added,  which  is 
followed  by  Problems  for  the  pupil  to  solve.  This  Book,  or  any 
part  of  it,  if  thoug'ht  best,  can  be  taken  immediately  after  com- 
pleting Book  111. 

The  Trigonometry  is  accompanied  by  the  necessary  Tables 
and  their  explanation,  and  presents  in  only  fifty-two  pages  all 
the  essential  principles  of  Plane  Trigonometry  given  by  both 
the  Geometrical  and  Analytical  methods,  and  so  arranged  that 
either  can  be  studied  independently  of*  the  other.  In  fourteen 
more  pages  is  given  the  application  of  those  principles  to  the 
measurement  of  heights  and  distances  and  the  determination 
of  areas. 

W.  F.  B. 

Cambridge,  Mass.,  April,  1872. 


CONTENTS. 


GEOMETRY. 

Paob 
Introductory  Definitions 1 


BOOK    I. 

Angles,  Lines,  Polygons 3 

Exercises        .       ' »         .         ,  22 

Ratio  and  Proportion ,        ,    25 

BOOK    II. 

Relations  of  Polygons ,        .        .    31 

Exercises *         -         .  45 

BOOK    III. 
The  Circle .49 

Exercises        . 61 

BOOK    IV. 

Geometry  of  Space. 

Planes  and  their  Angles        ,...,.,,     64 
Exercises 68 

BOOK    V. 
Polyedrons. 

Prisms,  Cylinders        .         .         .         .         .        .         .         .         .69 

Pyramids,  Cones 75 

The  Sphere 82 

Exercises 86 

BOOK    VI. 

Problems  of  Construction    , 89 

Exercises        . 106 


VI  CONTENTS. 


PLANE    TRIGONOMETRY. 

CHAPTER    I. 
Logarithms. 

Nature  of  Logarithms .1 

Explanation  of  Table  of  Logarithms        .....  3 

Multiplication  and  Division  by  Logarithms  .         .         .         .  .  7,  8 

Involution  and  Evolution  by  Logarithms         .         .         .        .  8,  9 

CHAPTER    II. 

Trigonometric  Functions.     Geometrical  Method. 

Definitions  of  Sine,  Tangent,  &c ,         .11 

Values  of  certain  Sines,  Tangents,  &c.     .         .         .  .         .  13 

Algebraic  Signs  of  the  Siiies,  Tangents,  &c.  .         .         .         .     14~ 

Explanation  of  Table  of  Sines,  Tangents,  &c.  .         .  ...       .         14 

CHAPTER    III. 

Solution  of  Plane  Triangles.    Geometrical  Method. 

Eight- Angled  Triangles       , .17 

Oblique- Angled  Triangles 22 

CHAPTER    IV. 

Trigonometric  Functions.    Analytical  Method. 

Definitions  of  Sine,  Tangent,  &c .29 

Values  of  Sines,  Tangents,  &c.         .....         .  32 

Algebraic  Signs  of  the  Sines,  Tangents,  &c 38 

CHAPTER    V. 

Solution  of  Plane  Triangles.     Analytical  Method. 

Right- Angled  Triangles 41 

Oblique- Angled  Triangles -.         .         46 

CHAPTER    VI. 

Practical  Applications. 

Heights  and  Distances        , 53 

Determination  of  Areas  ........  60 

Miscellaneous  Examples 64 


f^T 


PLANE    GEOMETRY 


^^.^t-iM--' ' '  ' 


INTRODlteTOBY-DEFINITIOiTS. 


1«   Mathematics  is  the  science 


of  qu^inti$5^ 


2.    Quantity  is  that  which  can  be  measured;    as  distance, 
time,  weight. 

3*   Geometry  is  that  branch  of  mathematics  which  treats  of 
the  properties  of  extension. 

4»   Extension   has  one   or   more  of  the   three    dimensions, 
length,  breadth,  or  thickness. 

5*    A  Point  has  position,  but  not  magnitude. 

6i    A  Line  has  length,  without  breadth  or  thickness. 

7     A  Straight  Line  is  one  whose  direction 
is  the  same  throughout ;  2i%  A  B. 

A  straight  line  has  two  directions  exactly  opposite,  of  which 
either  may  be  assumed  as  its  direction. 

The  word  line,  used  alone  in  this  book,  means  a  straight  line. 

8#    Corollary.     Two  points  of  a  line  determine  its  position. 

9.  A  Curved  Line  is  one  whose  direction     ^^ --^^^ 

is  constantly  changing;  as  CD. 

10.  A  Sur&x;e  has  length  and  breadth,  but  no  thickness. 

1 


2  PLANE   GEOMETRY. 

,■  Jl,    A  Plane ,i^  such  a  surface  that  a  straight  line  joining 

any  two'<>fH8  pewits  is  wholly  in  the  surface. 

' ;  J0.{  ^  Soiid  ha"^ .length,  breadth,  and  thickness. 

13*    Scholium.     The   boundaries   of  solids  are  surfaces;  of 
surfaces,  lines;  the  ends  of  lines  are  points. 

14.  A  Theorem  is  something  to  be  proved. 

15.  A  Problem  is  something  to  be  done. 

16.  A  Proposition  is  either  a  theorem  or  a  problem. 

17.  A  Corollary  is  an  inference  from  a  proposition  or  state- 
ment. 

18i    A  Scholium  is  a  remark  appended  to  a  proposition. 

19.  An  Hypothesis  is  a  supposition  in  the  statement  of  a 
proposition,  or  in  the  course  of  a  demonstration. 

20.  An  Axiom  is  a  self-evident  truth. 


AXIOMS. 

1.  If  equals  are  added  to  equals,  the  sums  are  equal. 

2.  If  equals  are  subtracted  from  equals,  the  remainders  are 
equal. 

3.  If  equals  are  multiplied  by  equals,  the  products  are  equal. 

4.  If  equals  are  divided  by  equals,  the  quotients  are  equal. 

5.  Like  powers  and  like  roots  of  equals  are  equal. 

G.    The  whole  of  a  magnitude  is  greater  than  any  of  its  parts. 

7.  The  whole  of  a  magnitude  is  equal  to  the  sum  of  all  its 
parts. 

8.  Magnitudes  respectively  equal  to  the  same  magnitude  are 
equal  to  each  other. 

9.  A  straight  line  is  the  shortest  distance  between  two  points. 


BOOK   I. 


ANGLES,  LINES,  POLYGONS. 

ANGLES. 
DEFINITIONS. 

1«   An  Angle  is  the  difference  in  direction  of  two  lines. 

If  the  lines  meet,  the  point  of  meeting,  B,  ^-^-^"^ 

is  called  the  vertex  ;  and  the  lines  A  Bj  B  C,  ^  .^^r^^___/7 
the  sides  of  the  angle. 

If  there  is  but  one  angle,  it  can  be  designated  by  the  letter 
at  its  vertex,  as  the  angle  B ;  but  when  a  number  of  angles 
have  the  same  vertex,  each  angle  is  designated  by  three  letters, 
the  middle  letter  showing  the  vertex,  and  the  other  two  with 
the  middle  letter  the  sides ;  as  the  angle  ABC, 


2.  If  a  straight  line  meets  another  so  as 
to  make  the  adjacent  angles  equal,  each 
of  these  angles  is  a  right  angle  ;  and  the  two 

lines  are  perpendicular  to  each  other.    Thus,        

AC  D  and  D  C  B,  being  equal,  are  right  an- 
gles, and  A  B  and  B  G  are  perpendicular  to  each  other, 

3.  An  Acute  Angle  is  less  than  a  right 
angle;  &&  U C B. 

4f    An  Obtuse  Angle  is  greater  than  a  right    A 
angle  ;  as  A  C  JS. 

Acute  and  obtuse  angles  are  called  oblique  angles. 


4  PLANE   GEOMETRV^. 

5t  The  Complement  of  an  angle  is  a  right  angle  minus  the 
given  angle.  Thus  (Fig.  in  Art.  7),  the  complement  oi  AG  D 
mACF—  ACD  =  DGF. 

6t  The  Supplement  of  an  angle  is  two  right  angles  minus 
the  given  angle.  Thus  (Fig.  Art.  7),  the  supplement  oi  AC  D 
is  {ACF-\-FCB)  —  AGI)z=DCB. 


THEOKEM  I. 

7»    The  sum  of  all  the  angles  formed  at  a  point  on  one  side  of 
a  straight  line^  in  the  same  2'dane,  is  equal  to  two  right  angles. 

Let  D  G  and  E  G  meet  the  straight 
line  A  B  2ii  the  point  G ;  then 
AGD  +  J)GF  +  FGB  =  two 
right  angles. 

At  G  erect  the  perpendicular,  GF; 
then  it  is  evident  that 

AGI)-\-DGF+FGB  =  AGD  +  I)GF+FGF  +  EGB 
=  AGF-i-FGB  =  two right  angles. 

8.  Corollary  1.     If  only  two  angles  are  ^ 
formed,  each  is  the  supplement  of  the  other. 
For  by  the  theorem, 

ACD  -{-D  C  B=itwQ  right  angles  ;  A ^ B 

therefore  AG  Dz=.  two  right  angles — D  GB^ 

or  D  C  B=z  two  right  angles —  A  CD. 

9.  Corollary  2.     The  sum  of  all  the  angles  formed  in  a 
plane  about  a  point  is  equal  to  four  right  angles. 

Let  the  angles  ^jS/),  DBE,  EBF, 
F  BGj  GB  A,  be  formed  in  the  same 
plane  about  the  point  B.  Produce 
A  B  -y  then  the  sum  of  the  angles 
above  the  line  A  C  is  equal  to  two  ' 
right  angles  ;  and  also,  the  sum  of 
the  angles  below  the  line  AC  ia  equal 


C 


BOOK  I. 


to  two  right  angles  (7)  * ;  therefore  the  sum  of  all  the  angles  at 
the  point  B  is  equal  to  four  right  angles. 

THEOREM  II. 

10»  If  at  a  point  in  a  straight  line  two  other  straight  lines 
upon  opposite  sides  of  it  make  the  sum  of  the  adjacent  angles  equal 
to  two  right  angles,  these  two  lines  form  a  straight  line. 

Let  the  straight  line  D  B  meet  the 
two  lines,  AB,  BG,  so  as  to  make 
ABD-\-DBC=^i^o  right  angles  : 
then  A  B  and  B  C  form  a  straight 
line.  ^ 

For  if  ^  ^  and  B  C  diO  not  form  a  straight  line,  draw  B  U  80 
that  A  B  and  B  E  shall  form  a  straight  line ;  then 

ABI)-\-DBB=  two  right  angles  (7) ; 
but  by  hypothesis, 

ABI)-\-I)BC=  two  right  angles  ; 
therefore  DBE  =  DBC 

the  part  equal  to  the  whole,  which  is  absurd  (Axiom  6) ;  there- 
fore AB  and  B  C  form  a  straight  line. 

THEOREM   III. 

11.  If  two  straight  lines  cut  each  other ^  the  opposite,  'or  vortical, 
angles  are  equal. 

Let  the  two  lines,  AB,    CD,  cut  each  other  at  E ;   then 

AEC  =  DEB. 
For  A  E  D  m  the  supplement  of  both    A 
AECaadBEB  (8)',  therefore 

AEC=DEB  0 

In  the  same  way  it  may  be  proved  that 

AED  —  CEB 

*  The  figures  alone  refer  to  an  article  in  the  same  Book  ;  in  referring  to 
an  article  in  another  Book  the  number  of  the  Book  is  prefixed. 


PLANE  GEOMETRY. 


THEOREM  IV. 


12.  Two  angles  whose  sides  have  the  same  or  opposite  directions 
are  equal. 

1st.  Let  BA  and  B  C^  including  the 
angle  B,  have  respectively  the  same  direc- 
tion as  ED  and  EF^  including  the  angle  E ; 
then  angle  B  =  angle  E. 

For  since  B  A  has  the  same  direction  as 
ED,  and  BC  the  same  as  EF,  the  differ- 
ence of  direction  of  B  A  and  B  C  must  be 
the  same  as  the  difference  of  direction  of  E  D  and  E  F;  that  is, 
angle  B  ==:  angle  E. 

2d.  Let  B  A  and  B  0,  including  the 
angle  B,  have  respectively  opposite  di- 
rections to  E  F  and  E  D,  including  the 
angle  E ;  then  angle  B  =  angle  E. 

Produce  I)  E  and  FE  so  as  to  form 
the  angle  G^^^;  then  (11) 

GEH—BEF 
and  GEHz=ABC  by  the  first  part  of  this 

proposition  ;  therefore  angle  B  z=l  angle  E. 

PARALLEL   LINES. 

13.  Definition.    Parallel  Lines  are  such  as    A B 

have  the  same  direction  \  Oi&AB  and  CD.  ^  ^ 


14.  Corollary.  Parallel  lines  can  never  meet.  For,  since 
parallel  lines  havG^  the  \ame  dir^ctk)n,-4f\  they  coincide^  at  one 
point,  they  would  coincide  throughout  and  form  one  and  the 
same  straipjht  line. 

Conversely,  straight  lines  in  the  same  plane  that  never  meet, 
however  far  produced,  are  parallel.  For  if  they  never  meet 
they  cannot  be  approaching  in  either  direction,  that  is,  they 
must  have  the  same  direction. 


BOOK  I.  7 

15,  Axiom.  Two  lines  parallel  to  a  third  are  parallel  to 
each  other. 

]6t  Definition.  When  parallel  lines  are  cut  by  a  third,  the 
angles  without  the  parallels  are  called 
external;  those  within,  internal ;  thus, 
AGE,  EGB,  CHF,  FEB  are  ex- 
ternal angles;  A  G H,  BGH,  G H C, 
G  H  D  are  internal  angles.  Two  in- 
ternal angles  on  the  same  side  of  the 
secant,  or  cutting  line,  are  called  internal  angles  on  the  same 
side  ;  2i?>  AG  H  smdi  G  H  C,  or  B  G  H  and  GHD.  Two  internal 
angles  on  opposite  sides  of  the  secant,  and  not  adjacent,  are 
called  alternate  internal  angles  ;  as  A  G H  ^n^  GHD,  ox  BGH 
2,ndiGHC. 

Two  angles,  one  external,  one  internal,  on  the  same  side  of 
the  secant,  and  not  adjacent,  are  called  opposite  external  and  in- 
ternal angles ;  2,^  E G  A  and  G H C,  or  E G B  and  GHD. 

THEOREM  V. 

17.    If  a  straight  line  cut  two  parallel  lines, 
1st.    The  opposite  external  and  internal  angles  are  equal. 
2d.    The  alternate  internal  angles  are  equal. 
3d.    Tlie  internal  angles  on  the  same  side  are  supplemembs  of 
each  other. 

Let  E  F  cut  the  two  parallels  A  B 

and  CD  ;  then  ^\^g 

1st.     The    opposite    external   and  ~ 


I 


internal   andes,   EGA    and   GHC,      ^  ^„ 

G — ■ ^^^v:; — D 

or    EGB    and    GHD,    are    equal,  \.^ 

since   their   sides    have   respectively 

the  same  directions  (12). 

2d.     The  alternate  internal  angles,   AGH  and  GHD,  or 

BGH  and  GHC,  are  equal,  since  their  sides  have  opposite 

directions  (12). 


8  PLANE   GEOMETRY. 

3d.  The  internal  angles  on  the  same  side,  ^6^^  and  G  H  G, 
OT  BG li  and  G H D,  are  supplements  of  each  other  ;  for  AGE 
is  the  supplement  of  A  G  E  (8),  which  has  just  been  proved 
equal  to  GIIG.  In  the  same  way  it  may  be  proved  that  BGU 
and  GHD  are  supplements  of  each  other. 


THEOREM  VI. 

CONVERSE   OP   THEOREM    V. 

11 8 1  If  d  Straight  line  cut  two  other  straight  lines  in  the  same 
plane,  these  two  lines  are  parallel, 

1st.    If  the  opposite  external  and  internal  angles  are  equal. 

2d.    If  the  alternate  internal  angles  are  equal. 

3d.  If  the  internal  angles  on  the  same  side  are  supplements  of 
each  other. 

Let  E  F  cut  the  two  lines  A  B  and 
GD  so  astomake^(?i?=  G H B, 
or  AG H  =  GHD,  or  BGII  and 
GUI)  supplements  of  each  other ; 
then  A  B  m  parallel  to  CD. 

For,  if  through  the  point  G  a  line 
is  drawn  parallel  to  CD,  it  will  make  the  opposite  external  and 
internal  angles  equal,  and  the  alternate  internal  angles  equal, 
and  the  internal  angles  on  the  same  side  supplements  of  each 
other  (17);  therefore  it  must  coincide  with  A  B -,  that  is,  AB 
is  parallel  to  CD. 


-x 


PLANE  FIGURES. 
DEFINITIONS. 


19".    A  Plane  Figure  is  a  portion  of  a  plane  bounded  by  lines 
either  straight  or  curved. 

When  the  bounding  lines  are  straight,  the  figure  is  a  polygon, 
and  the  sum  of  the  bounding  lines  is  the  perimeter. 


BOOK   I.  9 

20o    An  Equilateral  Polyg^on  is  one  whose  sides  are  equal 
each  to  each. 

21.  An  EquiaLgular  Polygon  is  one  whose  angles  are  equal 
each  to  each. 

22.  Polygons  whose  sides  are  respectively  equal  are  mutually 
equilateral. 

23.  Polygons  whose  angles  are  respectively  equal  are  mutu- 
ally equiangular. 

Two  equal  sides,  or  two  equal  angles,  one  in  each  polygon, 
similarly  situated,  are  called  homologous  sides,  or  angles. 

24.  Equal  Poljgons  are  those  which,  being  applied  to  each 
other,  exactly  coincide. 

25.  Of  Polygons,  the'  simplest  has  three  sides,  and  is  called 

a  triangle ;  one  of  four  sides  is  called  a  quadrilateral ;  one  of 
five,  a  pentagon  ;  one  of  six,  a  hexagon  ;  one  of  eight,  an  octagon  ; 
one  of  ten,  a  decagon. 


TRIAKGLES. 

26.   A  Scalene  Triangle  is  one  which  has 
no  two  of  its  sides  equal ;  as  J  -5  (7.  ^    ^ 


27.  An  Isosceles  Triangle  is  one  which  has  two 
of  its  sides  equal ;  SiS  I)  JE  F. 

DL \F 

G 

28.  An  Equilateral  Triangle  is  one  whose 
L-ides  are  all  equal ;  a.s  I G  IT, 


10 


PLANE  GEOMETRY. 


29t  A  Bight  Triangle  is  one  which  has  a 
right  angle  ;  SiS  JK  L. 

The  side  opposite  the  right  angle  is  called 
the  hypothenuse. 


30.   An  Obtuse-angled  Triangle  is   one 
which  has  an  obtuse  angle;  as  MNO.  j^-. 


31*  An  Acute-angled  Triangle  is  one  whose  angles  are  all 
acute;  as  D £J F. 

Acute  and  obtuse-angled  triangles  are  called  oblique-angled 
triangles. 

32 »  The  side  upon  which  any  polygon  is  supposed  to  stand 
is  generally  called  its  base;  but  in  an  isosceles  triangle,  as 
DjEF,  in  which  D  E  =  E  F,  the  third  side  D  F  i^ 
considered  the  base. 


THEOREM  yil. 
The  sum  of  the  angles  of  a  triangle  is  equal  to  two  right 


angles. 


Let  A  BChe  2i  triangle ;  the  sum 
of  its  three  angles,  A,  B,  C,  is  equal 
to  two  right  angles. 

Produce  A  G,  and  draw  CD  par- 
allel to  ^  ^ ;  then  J)CE=A,  be- 
ing external  internal  angles  (17); 
BG D  ■=.  B,  being  alternate  internal  angles  (17) ;  hence 

DGE-\-BGD-\-BGA=A-\-B-{-BGA 
but        DGE  -\-  BG D  -\-  BGA  —  two  right  angles  (7) ; 
therefore  A-\-B-\-BGA^=  two  right  angles. 

34.    Cm.  I.     If  two  angles  of  a  triangle  are  known,  the  thin! 
can  be  found  by  subtracting  their  sum  from  two  right  angles. 


BOOK   I.  11 

35 •  Cor.  2.  If  two  triangles  have  two  angles  of  the  one 
respectively  equal  to  two  angles  of  the  other,  the  remaining 
angles  are  equal. 

36*  Cor.  3.  In  a  triangle  there  can  be  but  one  right  angle, 
or  one  obtuse  angle. 

.  37.    Cor.  4.     In  a  right  triangle  the  sum  of  the  two  acute 
angles  is  equal  to  a  right  angle. 

38.  Cor.  5.  Each  angle  of  an  equiangular  triangle  is  equal 
to  one  third  of  two  right  angles,  or  two  thirds  of  one  right 
angle. 

39.  Cor.  6.     If  any  side  of  a  triangle  is  produced,  the  exte-    p 
rior  angle  is  equal  to  the  sum  of  the  two  interior  and  opposite. 


THEOREM  VIII. 

40.  If  two  triangles  have  two  sides  and  the  included  angle  of 
the  one  respectively  equal  to  two  sides  and  the  included  angle  of  the 
other y  the  two  triangles  are  equal  in  all  respects. 

In  the  triangles  ABC, 
DEF,  let  the  side  AB 
equal  DE,  AC  equal  DF, 
and  the  angle  A  equal  the 
angle  D  ;  then  the  triangle 
ABC  i%  equal  in  all  re- 
spects to  the  triangle  D  E  F. 

Place  the  side  A  B  on  its  equal  D  E^  with  the  point  A  on  the 
point  D,  the  point  B  will  be  on  the  point  E^  SiS  A  B  is  equal  to 
D  E ;  then,  as  the  angle  A  is  equal  to  the  angle  D,  A  C  will 
take  the  direction  D  F,  and  as  ^  (7  is  equal  to  D  F,  the  point 
C  will  be  on  the  point  F ;  and  BC  will  coincide  with  E  F, 
Therefore  the  two  triangles  coincide,  and  are  equal  in  all  re- 
spects. 


12 


PLANE  GEOMETRY. 


THEOREM  IX. 

41.  If  two  triangles  have  two  angles  and  the  included  side  of 
the  one  respectively  equal  to  two  angles  and  the  included  side 
of  the  other,  the  two  triangles  are  equal  in  all  respects. 

In  the  triangles  ABC 
and  D  E  F,  let  the  angle 
A  eqnal  the  angle  D,  the 
angle  C  equal  the  angle 
F,  and  the  side  A  C  equal 
D  F ;  then  the  triangle 
AB  C  m  equal  in  all  respects  to  the  triangle  DBF. 

Place  the  side  A  C  on  its  equal  D  F,  with  the  point  A  on  the 
point  D,  the  point  C  will  be  on  the  point  F^  ?i%  AC  is  equal  to 
D  F ;  then,  as  the  angle  A  is  equal  to  the  angle  D,  A  B  will 
take  the  direction  D  E ;  and  as  the  angle  C  is  equal  to  the 
angle  F,  CB  will  take  the  direction  FE ;  and  the  point  B  fall- 
ing at  once  in  each  of  the  lines  D  E  and  F  E  must  be  at  their 
point  of  intersection  E.  Therefore  the  two  triangles  coincide, 
and  are  equal  in  all  respects. 


THEOREM   X. 

42.  In  an  isosceles  triangle  the  angles  opposite  the  equal  sides 
are  equal. 

In   the  isosceles  triangle  A  B  C  let  B 

A  B  and  B  C  he  the  equal  sides  ;  then 
the  angle  A  is  equal  to  the  angle  C. 

Bisect  the  angle  A  B  C  hj  the  line 
B D ;  then  the  triangles  ABB  and 
BCD  are  equal,  since  they  have  the  two  sides  AB,  B D,  and 
the  included  angle  ABB  equal  respectively  to  BC,  B Z),  and 
the  included  angle  BBC  (40) ;  therefore  the  angle  A=:^  C. 

43.  Cor.  1.     From  the  equality  of  the  triangles  ABB  and 
BCD,  AD  =  DC,  and  the  angle  ADB  =  BDC;  that  is,  the 


BOOK   I.  13 

line  bisecting  the  angle  opposite  the  base  of  an  isosceles  triangle 
bisects  the  base  at  right  angles  and  also  bisects  the  triangle ; 
also  the  line  drawn  from  the  vertex  perpendicular  to  the  base 
of  an  isosceles  triangle  bisects  the  base,  the  vertical  angle,  and 
the  triangle.  And,  conversely,  the  perpendicular  bisecting  the 
base  of  an  isosceles  triangle  bisects  the  angle  opposite,  and  also 
the  triangle. 

44*    Cor.  2.     An  equilateral  triangle  is  equiangular. 

THEOREM  XI. 

45.  If  two  angles  of  a  triangle  are  equal,  the  sides  opposite  are 
also  equal. 

In  the  triangle  ABC  let  the  angle 
A  equal  the  angle  C ;  then  ^^  is  equal 

to^a 

Bisect  the  angle  A  B  C  hy  the  line 
B  D.      Now   by   hypothesis   the   angle    ~"      '  D 

A  is  equal  to  the  angle  C,  and  by  construction  the  angle  A  B  D 
is  equal  to  the  angle  D BC  ]  therefore  (35)  the  angle  A  D  B  m 
equal  to  the  angle  B D  C  ;  and  the  two  triangles  AB D,  D  BC, 
having  the  side  B  D  common  and  the  angles  including  B  D 
respectively  equal,  are  equal  (41)  in  all  respects  ;  therefore 
ABz=iBC. 

46.  Corollary.     An  equiangular  triangle  is  equilateral. 

THEOREM  XII. 

47«    The  greater  side  of  a  triangle  is  opposite  the  greater  angle ; 
and,  conversely,  the  greater  angle  is  opposite  the  greater  side. 

In  the  triangle  ^  i5  C  let  ^  be  greater 
than  C ;  then  the  side  AC  is  greater 
than  A  B. 

At  the  point  B  make  the  angle  CBD 
equal  to  the  angle  C ; 


14 


PLANE   GEOMETKY. 


then  (45)       DB=iDC 

and  A  C  =  AD  +  D  C=AD  +  DB 
But  (Axiom  9)  j^ 

AD+DByAB 


therefore 


AC^AB 


Conversely.  Let  A  C '^  A  B ;  then  the  angle  A  B  C  ^  C. 
Cut  oE  AD^=AB  and  join  B  D ;  then  as  A  D  =  A  B,  the 
angle  A  B  D  =  A  B  B  (42) ;  and  ADByC  (39) ;  therefore 
ABD-yC]  hut  ABCyABB;  therefore  A  B  0  y  G. 


y    ■  /y  THEOREM  XIII. 

48.    Two  triangles  mutually  equilateral  are  equal  in  all  respects. 

Let  the  triangle  A  BG 
have  A  B,  BG,  G  A  respec- 
tively equal  to  AD,  DG,  G A 
of  the  triangle  ABG \  then 
ABG  \^  equal  in  all  respects 
to  ADG. 

Place  the  triangle  ADG 
so  that  the  base  A  G  will  co- 
incide with  its  equal  A  G,  but  so  that  the  vertex  D  will  be 
on  the  side  of  A  (7,  opposite  to  B.  Join  B  D.  Since  by  hy- 
pothesis AB  z=  A  D,  AB D  is  an  isosceles  triangle ;  and  the 
angle  ABD  =  ADB  (42) ;  also,  since  BG  =  G D,  BG D  is 
an  isosceles  triangle ;  and  the  angle  D BG  =  G D B  ;  there- 
fore the  whole  angle  ABG=^ADG',  therefore  the  triangles 
ABG  and  ADG,  having  two  sides  and  the  included  angle  of 
the  one  equal  to  two  sides  and  the  included  angle  of  the  other, 
are  equal  (40). 


•  BOOK   I.  15 

49*    Scholium.     In  equal  triangles  the  equal  angles  are  oppo- 
site the  equal  sides. 


THEOREM  XIV. 

50«  Tivo  right  triangles  having  the  hypothenuse  and  a  side 
of  the  one  respectively  equal  to  the  hypothenuse  and  a  side  of  tie 
other  are  equal  in  all  respects. 

Let  A  B  G  have  the  hypothenuse  A  B 
and  the  side  B  C  equal  to  the  hypothe- 
nuse BD  and  the  side  BC  oi  BDC', 
then  are  the  two  triangles  equal  in  all 
respects. 

Place  the  triangle  BDC  %q>  that  the  side  B  G  will  coincide 
with  its  equal  BG^  then  G  D  will  be  in  the  same  straight  line 
with  A  G  (10).  An  isosceles  triangle  A  B  I)  is  thus  formed,  and 
B  G  being  perpendicular  to  the  base  divides  the  triangle  into 
the  two  equal  triangles  ABG  and  B D  G  (43). 


THEOREM  XV. 

51.    If  from  a  point  without  a  straight  line  a  perpendicular 

and  oblique  lines  he  drawn  to  this  line, 

1st.    The  perpendicular  is  shorter  than  any  oblique  line. 

2d.    Any  two  oblique  lines  equally  distant  from  the  perpendicu- 
lar are  equal. 

3d.    Of  two  oblique  lines  the  more  remote  is  the  greater. 

Let  A  be  the  given  point,  BG  the 
given  line,  A  D  the  perpendicular,  and 
AE,  AB,  AG  oblique  lines.  B^^ 

Ist.    In  the  triangle  A  D  E,  the  an-  ^ 

gle  A  D  E  being  a  right  angle  is'greater  than  the  angle  A  E  D; 
therefore  A  D  <i  A  E  (47). 


R       D 


16  PLA.NE   GEOMETRY. 

2d.  J{  DH—DC;  then  the  two 
triangles  ABE  and  ADC,  having  two 
sides  A  D,  D  E,  and  the  included  angle 
A  D  E  respectively  equal  to  the  two 
sides  AD,  DC,  and  the  included  angle  ADC,  are  equal  (40), 
and  ^  jE'  is  equal  to  A  C. 

3d.  UDB^D  E;  then,  &,s  A  D  E  is  fx  right  angle,  AED 
is  acute ;  hence  A  E  B  is,  obtuse,  and  must  therefore  be  greater 
than  ABE  (36) ;  hence  AByAE  (47). 

52i  Corollary.  Two  equal  oblique  lines  are  equally  distant 
from  the  perpendicular. 


THEOREM  XVI. 

53  •  If  cit  the  middle  of  a  straight  line  a  perpendicular  is 
draivn, 

1st.  Any  point  in  the  perpendicular  is  equally  distant  from  the 
extremities  of  the  line. 

2d.  Any  point  without  the  perpendicular  is  unequally  distant 
from  the  same  extremities. 

Let  CD  he  the  perpendicular  at  the  middle  E 

of  the  line  A  B ;  then  ^(  \ 

1st.    Let  D  be  any  point  in  the  perpendicu- 
lar; draw  i>^  and  D B.     Since  CA  =  C B,    ^/ 
D  A  =zDB  (51). 

2d.  Let  E  be  any  point  without  the  perpen- 
dicular ;  draw  EA  and  EB,  and  from  the  point 
D,  where  E  A  cuts  D  C,  draw  D  B.     The  an- 


gle ABE^ABD  =  BAD;  hence,  in  the  triangle  AEB 
since  the  angle  A  B E  :>  B AE,  E A  y  E B  (47). 


BOOK    I. 


17 


QUADRILATERALS. 
DEFmmONS. 


54.    A  TrapeziTim  is  a  quadrilateral  which 
has  no  two  of  its  sides'  parallel ;  as  A  B  C  D. 


55.    A    Trapezoid    is    a    quadrilateral  ^        ^ ^  .;f 

which  has  only  two  of  its  sides  parallel ; 
mJSFGff.  H' 


\ 


56*    A  Farallelogpram  is  a  quadrilateral  whose  opposite  sides 
are  parallel ;  as  IJ K L,  or  MN  0  P,  or  Q  R  S  T,  ov  U  V  W X. 

I J 

57.    A  Bectangle  is  a  rij^ht-angled  parallel- 
ogram ;  3iS  UK L. 


58.    A  Sqnare  is  an  equilateral  rectangle ; 
asMJVOF. 


59.    A  Rhomboid  is  an  oblique-angled  par- 
allelogram ;  as  Q  E  S  T. 


60.    A  Rhombus  is  an  equilateral  rhomboid ; 
aaUVWX. 


Q  T 

u  r 


J 


X  w 

61.    A  Diagonal  is  a  line  joining  the  vertices  of  two  angles 
not  adjacent ;  as>  D  B. 


18  PLANE   GEOMETRY. 


THEOREM  XVII. 


62.   In  a  parallelogram  the  opposite  sides  are  equal,  and  the 
opposite  angles  are  equal. 

Let  ABC D  he  a  parallelogram ;  then  B C 

will  AB  =  DC,BC  —  AD,  the  angle         /^T  7 

A=zCraxidiB  =  D.  I ^"'^^^...   / 

Draw  the  diagonal  B  D.     As  B  C  and      A  D 

A  D  are  parallel,  the  alternate  angles  C  B  D  and  B  D  A  are 
equal  (17);  and  as  ^  ^  and  DC  are  parallel,  the  alternate 
angles  AB  D  and  B  D  C  Sive  equal ;  therefore  the  two  triangles 
A  B  D  and  B  D  C,  having  the  two  angles  equal,  and  the  in- 
cluded side  ^i>  common,  are  equal  (41) ;  and  the  sides  opposite 
the  equal  angles  are  equal,  viz.  :  A  B  =z  D  C  and  BC  =^  A  D  ] 
also  the  angle  A  =z  C,  and  the  angle 

ABC=ABD  +  DBC  =  BDC-\-BDA=ADC 

63*    Cor.  1.     The  diagonal  divides  a  parallelogram  into  two 
equal  triangles. 

64*    Cor.  2.     Parallels  included  between  parallels  are  equal. 

THEOREM  XVIII. 

65.    If  two  sides  of  a  quadrilateral  are  equal  and  parallel,  the 
figure  is  a  parallelogram. 

Let  A  B  CD  be  a  quadrilateral  having  B O 

B  C  equal    and    parallel   to  A  D  ;   then  /  ''"^>.,^  / 

AB  CD  is  a  parallelogram.  /  ^'^ -.,   / 

Draw  the  diagonal  B  D.     As  B  C  m      A  D 

parallel  to  AD,  the  alternate  angles  CBD  and  BDA  are  equal 
(17);  therefore  the  two  triangles  CBD  and  BDA,  having 
the  two  sides  C B,  B D,  and  the  included  angle  CBD  respec- 
tively equal  to  the  two  sides  AD,  D  B,  and  the  included  angle 
A  D  B,  are  equal  (40),  and  the  alternate  angles  A  B  D  and 
B  D  C  are  equal ;  therefore  A  B  is  parallel  to  Z>  C  (18),  and 
A  B  C  D  is  a  parallelogram. 


BOOK   I.  1-9 

THEOREM   XIX. 

66*  The  line  joining  the  middle  points  of  the  two  sides  of  a 
trapezoid  which  are  not  parallel  is  parallel  to  the  two  parallel 
sides,  and  equal  to  half  their  sum. 

Let  ^i^  join  the  middle  points  of  the  sides  AB  and  CD, 
which  are  not  parallel,  of  the  trapezoid  A  B  C  D  ;  then 

1st.    EF  is  parallel  to  BC  and  AD.    j^ G      C 

Through  F  draw  G  H  parallel  to  ^  ^, 
meeting  AD  produced  in  H.     The  an-      E^ 
gles  GFC  and  DFR  are  equal  (11); 


also  the  angles  G  C  F  and  FD II  (17)  ;  D      H 

and  the  side  G  F  \s,  equal  to  FD ;  therefore  the  triangles  GFC 
and  D  F  H  are  equal  (41),  and 

GF=FII=z\GH 

But  ^  ABG H\^a  parallelogram,  GIIz=BA  (62) ;  therefore 

FII==iBAz=AE 

therefore  AE F H  is  a  parallelogram  (65),  and  E F is  parallel 
to  A  D,  and  therefore  also  to  B  C. 

2d.  EF=i(AD  +  BC) 

For  as  AEFH  and  EBGF  are  parallelograms 

EF=AH=AD+Dff 
and  also  EF—BG=BC—G.C 

Now,  as  the  two  triangles  GFC  and  D  F  H  are  equal, 
GC  =  D  H ;  therefore,  if  we  add  the  two  equations,  we 
shall  have 

2  EF=AD-\-  BC 

or  EF=i(AD  +  BC) 


E 


20  PLANE  GEOMETRY. 

THEOREM  XX. 

67 »    Th£  sum  of  the  interior  angles  of  a  polygon  is  equal  to 
tvrlce  as  many  right  angles  as  it  has  sides  minus  two. 

Let  ABC D E F  be  the  given  polygon  ;  q 

the  sum  of  all  the  interior  angles  A,  B,  C^  /^'^^^^^^^'^\ 

D,  E,  F,  is  equal  to  twice  as  many  right     j^:''l -^D 

angles  as  the  figure  has  sides  minus  two.  y  "~^-,.,  / 

For  if  from  any  vertex  A,  diagonals  A  (7,  V 

AD,  AE,  are  drawn,  the  polygon  will  be 
divided  into  as  many  triangles  as  it  has  sides  minus  two  ;  and 
the  sum  of  the  angles  of  each  triangle  is  equal  to  two  right 
angles  (33) ;  therefore  the  sum  of  the  angles  of  all  the  triangles, 
that  is,  the  sum  of  the  interior  angles  of  the  polygon,  is  equal  to 
twice  as  many  right  angles  as  the  polygon  has  sides  minus  two. 


PRACTICAL  QUESTIONS. 


1.  Do  two  lines  that  do  not  meet  form  an  angle  with  each  other  ?  Two 
lines  not  in  the  same  plane  ? 

2.  Does  the  magnitude  of  an  angle  depend  upon  the  length  of  its  sides  ? 

3.  If  a  right  angle  is  90°,  what  is  the  complement  of  an  angle  of  27°  ? 
of  51°  ?  of  91°  ?  of  153°  ?  What  is  the  supplement  of  an  angle  of  13°  ? 
of  83°  ?  of  97°  ?  of  217°  ? 

4.  If  three  of  four  angles  formed  at  a  point  on  the  same  side  of  a  straight 
line,  in  the  same  plane,  contain  respectively  15°,  27°,  and  99°,  how  many 
degrees  does  the  fourth  angle  contain  ? 

5.  If  five  of  six  angles  formed  in  a  plane  about  a  point  are  respectively 
11°,  53°,  74°,  19°,  and  117°,  how  many  degrees  are  there  in  the  sixth  angle  ? 

6.  On  opposite  sides  of  a  line  A  B  are  two  lines  making  with  A  B,  at 
the  point  A,  the  first  an  angle  of  29°,  and  the  second  an  angle  of  61°  ;  how 
are  these  two  lines  related  ? 


BOOK   I.  21 


7.  Can  two  polygons,  each  not  equilateral,  be  mutually  equilateral  ? 

8.  Can  two  polygons,  each  not  equiangular,  be  mutually  equiangular 


9.  If  two  angles  of  a  triangle  are  respectively  32°  and  43°,  how  many 
degrees  are  there  in  the  remaining  angle  ? 

10.  If  one  acute  angle  of  a  right  triangle  is  24°,  how  many  degrees  rre 
there  in  the  other  acute  angle  ? 

11.  How  many  degrees  in  each  angle  of  an  equiangular  triangle  ? 

12.  How  many  degrees  in  each  angle  at  the  base  of  an  isosceles  triangle 
whose  vertical  angle  is  14°  ? 

13.  How  many  degrees  in  each  acute  angle  of  a  right-angled  isosceles 
triangle  ? 

14.  If  one  of  the  angles  at  the  base  of  an  isosceles  triangle  is  double 
the  angle  at  the  vertex,  how  many  degrees  in  each  ? 

15.  If  the  angle  at  the  vertex  of  an  isosceles  triangle  is  double  one  of 
the  angles  at  the  base,  how  many  degrees  in  each  i 

16.  Two  triangles  mutually  equilateral  are  mutually  equiangular  (48). 
Are  two  triangles  mutually  equiangular  also  mutually  equilateral  ? 

17.  Is  a  square  a  parallelogram  ?     Is  a  parallelogram  a  square  ? 

18.  Is  a  rectangle  a  parallelogram  ?     Is  a  parallelogi-am  a  rectangle  ? 

19.  How  many  sides  equal  to  one  another  can  there  be  in  a  trapezoid  ? 
How  many  in  a  trapezium  ? 

20.  How  many  degrees  in  each  angle  of  an  equiangular  pentagon  ?  an 
equiangular  hexagon  ?  octagon  ?  decagon  ?  dodecagon  ? 

21.  If  the  parallel  sides  of  a  trapezoid  are  respectively  8  feet  and  13  feet 
in  length,  how  long  is  the  line.joinmg  the  middle  points  of  the  other  two 
sides  ? 

22.  If  one  of  the  angles  of  a  parallelogram  is  120°,  how  many 
are  there  in  each  of  the  other  angles  ? 


22  PLANE   GEOMETRY. 


EXERCISES. 

The  following  Theorems,  depending  for  their  demonstration  upon 
those  already  demonstrated,  are  introduced  as  exercises  for  the  pupil. 
In  some  of  them  references  are  made  to  the  propositions  upon  which 
the  demonstration  depends.  They  are  not  connected  with  the  prop- 
ositions in  the  following  books,  and  can  be  omitted  if  thought  best. 

68i  Two  angles  whose  sides  have,  one  pair  the  same,  the  other 
opposite  directions,  are  supplements  of  each  other.     (12.)     (8.) 

69«  Any  side  of  a  triangle  is  less  than  the  sum,  but  greater  than 
the  difference,  of  the  other  two.     (Axiom  9.) 

70.  The  sum  of  the  lines  drawn  from  a  point  within  a  triangle  to 
the  extremities  of  one  of  the  sides  is  less  than  the  sum  of  the  other 
two  sides. 

Produce  one  of  the  lines  to  the  side  of  the  triangle.     (Axiom  9.) 

71  •  The  angle  included  by  the  lines  drawn  from  a  point  within  a 
triangle  to  the  extremities  of  one  of  the  sides  is  greater  than  the 
angle  included  by  the  other  two  sides. 

Produce  as  in  (70).     (39.) 

72.  The  angle  at  the  base  of  an  isosceles  triangle  being  one  fourth 
of  the  angle  at  the  vertex,  if  a  perpendicular  is  drawn  to  the  base  at 
its  extreme  point  meeting  the  opposite  side  produced,  the  triangle 
formed  by  the  perpendicular,  the  side  produced,  and  the  remaining 
side  of  the  triangle,  is  equilateral. 

73.  If  an  isosceles  and  an  equilateral  triangle  have  the  same  base, 
and  if  the  vertex  of  the  inner  triangle  is  equally  distant  from  the  ver- 
tex of  the  outer  and  the  extremities  of  the  base,  then  the  angle  at 
the  base  of  the  isosceles  triangle  is  J  or  j  of  its  vertical  angle,  accord- 
ing as  it  is  the  inner  or  the  outer  triangle. 

74.  Prove  Theorem  VII.  by  first  drawing  a  line  through  B  par- 
allel to  A  C. 

75.  Prove  Theorem  VII.  by  drawing  a  triangle  upon  the  floor, 
walking  over  its  perimeter,  and  turning  at  each  vertex  through  an 
angle  equal  to  the  angle  at  that  vertex. 


BOOK   I.  23 

76.  Only  one  perpendicular  to  a  straight  line  can  be  drawn  from 
a  point. 

(Two  cases.  1st.  When  the  point  is  without  the  line.  2d.  When 
the  point  is  within  the  line.) 

77«    Two  straight  lines  perpendicular  to  a  third  are  parallel.    (13.) 

78.  If  a  line  joining  two  parallels  is  bisected,  any  other  line  drawn 
through  the  point  of  bisection  and  joining  the  parallels  is  bisected. 

79.  If  two  triangles  have  two  sides  of  one 
respectively  equal  to  two  sides  of  the  other,  but 
the  included  angles  unequal,  the  third  side  of  the 
one  having  the  included  angle  greater  is  greater 
than  the  third  side  of  the  other. 

Place  the  triangles  as  in  the  figure;  draw  BE 
bisecting  the  angle  C  BD,  and  join  C  and  E. 

80.  (Converse  of  79.)  If  two  triangles  have  two  sides  of  one 
respectively  equal  to  two  sides  of  the  other,  but  the  third  sides  un- 
equal, the  included  angle  of  the  one  having  the  third  side  greater  is 
greater  than  the  included  angle  of  the  other. 

(Prove  it  by  proving  any  other  supposition  absurd.) 

81.  Prove  in  Theorem  XIII.  the  angles  of  the  two  triangles 
equal  by  reference  to  (80)  then  that  the  triangles  are  equal  by  (40) 
or  (41). 

82.  (Converse  of  part  of  62.)  If  the  opposite  sides  of  a  quad- 
rilateral are  equal,  the  figure  is  a  parallelogram. 

83.  (Converse  of  part  of  62.)  If  the  opposite  angles  of  a  quadri- 
lateral are  equal,  the  figure  is  a  parallelogram. 

84.  (Converse  of  63.)  If  a  diagonal  divides  a  quadrilateral  into 
two  equal  triangles,  is  the  figure  necessarily  a  parallelogram  ? 

85.  The  diagonals  of  a  parallelogram  bisect  each  other. 

86.  (Converse  of  85.)  If  the  diagonals  of  a  quadrilateral  bisect 
each  other,  the  figure  is  a  parallelogram. 

87.  The  diagonals  of  a  rhombus  bisect  each  other  at  right  angles. 


24  PLANE  GEOMETRY. 

88.  (Converse  of  87.)  If  the  diagonals  of  a  quadrilateral  bisect 
each  other  at  right  angles,  the  figure  is  a  rhombus  or  a  square. 

89.  The  diagonals  of  a  rectangle  are  equal. 

90.  The  diagonals  of  a  rhombus  bisect  the  angles  of  the  rhombus. 

91 .  Straight  lines  bisecting  the  adjacent  angles  of  a  parallelogram 
are  perpendicular  to  each  other. 

92.  From  the  vertices  of  a  parallelogram  measure  equal  distances 
upon  the  sides  in  order.  The  lines  joining  these  points  on  the  sides 
form  a  parallelogram. 

93.  Prove  Theorem  XX.  by  joining  any  point  within  to  the  ver- 
tices of  the  polygon. 


94.  If  the'  sides  of  a  polygon,  as 
A  B  OD  EF,  are  produced,  the  sum  of 
the  angles  a,  &,  c,  c?,  e,  /,  is  equal  to  four 
right  angles. 


95.  If  a  pavement  is  to  be  laid  with  blocks  of  the  same  regular 
form,  that  is,  blocks  whose  faces  are  equiangular  and  equilateral, 
prove  that  their  upper  faces  must  be  equilateral  triangles,  S(|uares,  or 
hexagons.     (67  ;  9.) 

96.  If  two  kinds  of  regular  figures,  with  sides  of  the  same  length, 
are  to  be  used  at  each  angular  point,  show  that  the  pavement  can  be 
laid  only  with  blocks  whose  upper  faces  are, 

1st.   Triangles  and  squares. 
2d.    Triangles  and  hexagons. 
3d.   Triangles  and  dodecagons. 
4th.    Squares  and  octagons. 
IIow  many  of  each  must  there  be  at  each  angular  point? 

97.  If  three  kinds  of  regular  figures,  with  sides  of  the  same 
length,  are  to  be  used  at  each  angular  point,  show  that  the  pavement 
can  be  laid  only  with  blocks  whose  upper  faces  are, 

1st.   Triangles,  squares,  and  hexagons. 
2d.   Squares,  hexagons,  and  dodecagons. 
How  many  of  each  must  there  be  at  each  angular  point  ? 


RATIO   AND   PEOPOETIOK 

DEFINITIONS. 

(It  is  necessary  to  understand  the  elementaiy  principles  of  ratio  and  pro- 
portion before  entering  upon  the  Books  that  are  to  follow.  It,  is  therefore 
introduced  here,  but  not  numbered  as  one  of  the  Books  of  Geometry,  as  it 
belongs  properly  to  Algebra.  Reference  to  the  propositions  in  ratio  and 
proportion  will  be  made  by  the  abbreviation  Pn.,  with  the  number  of  the 
article  annexed.) 

1*  Ratio  is  the  relation  of  one  quantity  k>  another  of  the 
same  kind ;  or  it  is  the  quotient  which  arises  from  dividing  one 
quantity  by  another  of  the  same  kind. 

Ratio  is  indicated  by  writing  the  two  quantities  after  one  an- 
other with  two  dots  between,  or  by  expressing  the  division  in 
the  form  of  a  fraction.  Thus,  the  ratio  of  a  to  6  is  written, 
a  :  6,  or  - ;  read,  a  is  to  b,  or  a  divided  by  b. 

2i  The  Terms  of  a  ratio  are  the  quantities  compared,  whether 
simple  or  compound. 

The  first  term  of  a  ratio  is  called  the  antecedent,  the  other 
the  consequent ;  the  two  terms  together  are  called  a  couplet. 

3t    An  Inverse  or  Reciprocal  Ratio  of  any  two  quantities  is 

the  ratio  of  their  reciprocals.     Thus,  the  direct  ratio  of  a  to  6 

is  a  :  ft,  that  is,  - ;  the  inverse  ratio  of  a  to  6  is  -  :  -,   that  is, 

1        1        &         , 
-  -i-  T  =  -  or  6  :  a. 
aba 

4i    Proportion  is  an  equality  of  ratios.     Four  quantities  are 
in  proportion  when  the  ratio  of  the  first  to  the  second  is  equal 
to  the  ratio  of  the  third  to  the  fourth. 
2 


26  PLANE  GEOMETRY. 

The  equality  of  two  ratios  is  indicated  by  the  sign  of  equality 
(=),  or  by  four  dots  (:  :). 

Thus,  a  :  5  =  c  :  c?,  or  a  :  6  :  :  c  :  o?,  or  -  =  - ;  read  a  to  6 
equals  c  to  c?,  or  a  is  to  6  as  c  is  to  d,  or  a  divided  by  h  equals  c 
divided  by  d. 

5.  In  a  proportion  the  antecedents  and  consequents  of  the 
two  ratios  are  respectively  the  antecedents  and  consequents  of  the 
proportion.  The  first  and  fourth  terms  are  called  the  extremeSy 
and  the  second  and  third  the  means. 

6i  When  three  quantities  are  in  proportion,  e.  g.  a  :  6  =  ft  :  c, 
the  second  is  called  a  mean  proportional  between  the  other  two ; 
and  the  third,  a  third  proportional  to  the  first  and  second. 

7i  A  proportion  is  transformed  by  Alternation  when  antece- 
dent is  compared  with  antecedent,  and  consequent  with  conse- 
quent. 

8.  A  proportion  is  transformed  by  Inversion  when  the  ante- 
cedents are  made  consequents,  and  the  consequents  antece- 
dents. 

9t  A  proportion  is  transformed  by  Composition  when  in  each 
couplet  the  sum  of  the  antecedent  and  consequent  is  compared 
with  the  antecedent  or  with  the  consequent. 

lOi  A  proportion  is  transformed  by  Division  when  in  each 
couplet  the  difference  of  the  antecedent  and  consequent  is  com- 
pared with  the  antecedent  or  with  the  consequent. 

11,  Axiom.  Two  ratios  respectively  equal  to  a  third  are 
equal  to  each  other. 


RATIO   AND    PROPORTION.  27 

THEOREM   I. 

12f  In  a  proportion  the  product  of  the  extremes  is  equal  tc 
the  product  of  the  means. 

Let  a  :  h  =  c  :  d 

that  is  -  =  - 

b   ^     d 

Cleariug  of  fractions  ad=z  be 

13*  Scholium.  A  proportion  is  an  equation;  and  making 
the  product  of  the  extremes  equal  to  the  product  of  the  means 
is  merely  clearing  the  equation  of  fractions. 


THEOREM  II. 

14.  If  the  product  of  two  quantities  is  equal  to  the  product  of 
two  others,  the  factors  of  either  product  may  he  made  the  extremes, 
and  the  factors  of  the  other  the  means  of  a  proportion. 


Let 


a  =  he 


Dividing  hy  hd  h^^  1 

that  is  a  :  b  =  c  :  d 


THEOREM  in. 

15.    If  four  quantities  are  in  proportion,  they  loill  he  in  pro- 
portion hy  alternation. 

Let  a  '.h  =^  c  :  d 

By  (12)  ad  =  bc 

By  (14)  a:c  =  h:d 


28  PLANE   GEOMETRY. 

THEOREM  IV. 

16i    If  four  quantities  are  in  proportion^  they  will  he  in  pro- 
portion by  inversion. 

Let  a  :  h  =  c  :  d 

By  (12)  ad  =  bc 

By  (14)  .  b  -.a^d'.c 

THEOREM  V. 

17«    If  four  quantities  are  in  proportion^  they  will  he  in  pro- 
portion by  composition. 

Let  a  :  b  ==  c  :  d 

that  is 

Adding  1  to  each  member 

or 
that  is 


THEOREM  VI. 

18i    If  four  quantities  are  in  proportion,  they  will  he  in  pro- 
portion by  division. 

Let  a  -.h  ■=^  c  \  d 

that  is 


b 

'  d 

1  +  ^- 

■l^' 

a-\-b 
b      ~ 

c-^d 
■       d 

a  +  b  :b-=: 

c  +  d\ 

:d 

Subtracting  1  from  each  member 


or 
that  is 


a c 

b~  d 

1-^-2-' 

a  —  b        c  —  d 
b      ~      d 

a  —  h  :  h  =2  c  —  d 

\d 

RATIO   AND   PROPORTION.  29 

19.    Corollary.     From  (17)  and  (18),  by  means  of  (15)  and 
(H), 

If  a  :h  =z  c  :  d 

then  a-^-b  :  a  —  b=:c  -{-d  :c  —  d 


THEOREM  VII. 

20»  Equimidtiples  of  two  quantities  have  tlie  same  ratio  as  the 
quantities  themselves. 

T,  a        ma 

b        mb 
that  is  a  :b  =.  ma  :  mb 

21  •  Corollary.  It  follows  that  either  couplet  of  a  proportion 
may  be  multiplied  or  divided  by  any  quantity,  and  the  result- 
ing quantities  will  be  in  proportion.  And  since  by  (15),  if 
a  :b  =z  ma  :  mb,  a  :  m  a=b  :  mb  or  ma  :  a  =  mb  :  b,  it 
follows  that  both  consequents,  or  both  antecedents,  may  be 
multiplied  or  divided  by  any  quantity,  and  the  resulting  quan- 
tities will  be  in  proportion. 


THEOREM  VIII. 

22  •    If  four  quantities  are  in  proportion,  like  powers  or  like 
roots  of  these  quantities  will  be  in  proportion. 

Let  a  '.b  -~  c  '.  d 

au  X  •  a        c 

that  is  T  =  -; 

0        d 

Hence  -j-  =  -- 

5«        d^ 

that  is  cr*»  :  6"  =  c**  :  c?" 

Since  n  may  be  either  integral  or  fractional,  the  theorem  is 
proved. 


30  PLANE   GEOMETKY. 

THEOREM   IX. 

23.  If  any  number  of  quantities  are  proportional^  any  antece- 
dent is  to  its  consequent  as  the  sum  of  all  the  antecedents  is  to  the 
stem  of  all  the  consequents. 


Let 

a  :b^=c  :  d=^e  :f 

Now 

ab  =  ab                            (A) 

and  by  (12) 

ad^bc                             (B) 

and  also 

,     a/=6e                              (C) 

Adding  (A),  (B),  (C) 

a{b  + 

^+/)=6(a+c  +  e) 

Hence,  by  (14) 

a:6=a  +  c  +  6:6  +  6?+/ 

' 

THEOREM  X. 

24  •    //'  there  are  two  sets  of  quantities  in  proportion,  their  pro- 
ducts, or  quotients,  term  by  term,  will  be  in  proportion. 

Let  a  :  b  =z  c  :  d 

and                    ,  e\f=^g:h 

By  (12)  ad  =  bc  (A) 

and  ehz=fg  (B) 

Multiplying  (A)  by  (B)  adeh  =  b cfg  (C) 

Dividing  (A)  by  (B)  ^=)^  (D) 

From  (C)  by  (U)  ae\bf=cg\dh 

1  «         /T^x  abed 

and  from  (D)  -  :  -  =  -  :  7 

^    '  t    f       g     h 


BOOK   II 


EELATIONS   OF  POLYGONS. 


DEFINITIONS. 

1,  The  Area  of  a  polygon  is  the  measure  of  its  surface.  It 
is  expressed  in  units,  which  represent  the  number  of  times  the 
polygon  contains  the  square  unit  that  is  taken  as  a  standard. 

2.  Equivalent  Polygons  are  those  which  have  the  same  area. 

B 
3*.  The  Altitude  of  a  triangle  is  the  perpendic- 
ular distance  from  the  opposite  vertex  to  the  base, 
or  to  the  base  produced ;  as  B  D. 


it  The  Altitude  of  a  parallelogram  is  the 
perpendicular  distance  from  the  opposite  side 
to  the  base  ;  as  IK. 


5(  The  Altitude  of  a  trapezoid  is  the 
perpendicular  distance  between  its  paral- 
lel sides ;  as  P  /?. 


'    E  K  H 

M      P     N 


THEOREM  I. 


6*    Two  polygons  mutually  equiangular  and  equilateral  are 
equal. 


Let  ABGDE F  and 
GHIKLM  be  two  poly- 
gons  having    the    sides    A 
AB,BC,  CD,DE,EF, 
FA  and  the  angles  A,  B, 


D    a 


PLANE   GEOMETEY. 


(7,  D,  E,  F  of  the  one  re- 
spectively equal  to  the 
sides  G II,  R  I,  I K,  K  L, 
L  M,  M  G,  and  the  angles 
G,  H,  /,  K,  L,  M  of  the 
other  j  then  is  the  poly- 
gon ABGD  EF  equal  to  the  polygon  Gil  I  XL  M. 

For  if  the  polygon  ABG D E F  is  applied  to  the  polygon 
GHIKLM  so  that  A  B  shall  he  on  G H  with  the  point  A  on 
G,  B  will  fall  on  H,  as  A  B  and  G  H  are  equal ;  and  as  the 
angle  B  is  equal  to  the  angle  //,  B  G  will  take  the  direction 
HI;  and  as  BG  is  equal  to  ///,  the  point  G  will  fall  on  I\ 
and  so  also  the  points  D,  E,  F  will  fall  on  the  points  K,  L,  M\ 
and  the  polygon  ABG D E F  will  coincide  with  the  polygon 
GHIKLM,  and  therefore  be  equal  to  it. 


THEOREM  II. 

7i    The  area  of  a  rectangle  is  equal  to  the  product  of  its  basA 
and  altitude. 


A    H    I    J    K   D 


Let  ABGD  be  a  rectangle ;  its  area      B    0    P    Q    K    C 
=zAD  X  AB. 

Suppose  A  B  and  A  D  to  he  divided 
into  any  number  of  equal  parts,  A  E, 
EF,  AH,  HI,  &c.,  and  through  the 
points  of  division,  lines  EL,  F M,  HO, 
IP,  &c.  be  drawn  parallel  to  the  sides  of 
the  rectangle ;  then  the  rectangle  will  be  divided  into  squares ; 
these  squares  will  be  equal  to  each  other  (6).  If  one  of  the 
equal  parts,  A  E,  represents  the  linear  unit,  then  one  of  the 
squares,  A  E  S  H,  represents  the  square  unit ;  and  there  will  be 
as  many  square  units  in  the  rectangle  A  E  L  B  as  there  are 
linear  units  in  A  I)  -,  and  as  many  square  units  in  the  rectangle 
ABGD  as  there  are  square,  units  in  AE  LD  multiplied  by  the 
number  representing  the  number  of  linear  units  in  A  B  i  that 


BOOK  II.  33 

is,  the  area  of  the  rectangle  is  equal  to  the  product  of  its  base 
and  altitude,  that  is  =  A  B  X  AB. 

8.  Scholium.  If  A  I)  and  A  B  have  no  common  measure, 
the  linear  unit  may  be  taken  as  small  as  we  please,  that  is,  so 
small  that  the  remainders  will  be  infinitesimal,  and  can  be  neg- 
lected. 

9.  Corollary.  The  area  of  a  square  is  the  square  of  one  of 
its  sides. 

THEOREM  III. 

10.  The  area  of  a  parallelogram  is  equal  to  the  product  of  its 
base  and  altitude. 

Let  D  F  hQ  the  altitude  of  the  paral-     E   B F    0 

lelogram   A  BG D \    then    the    area    of 
ABGDz=AD  X  DF. 


At  A  draw  the  perpendicular  A  E  meet-     A  D 

ing  C B  produced  in  E ]  AEFD  is  a  rectangle  equivalent  to 
the  parallelogram  ABC  D.  For  the  two  triangles  AE  B  and 
DFC,  having  the  sides  A  E,  A  B  equal  respectively  to  the 
sides  D  Fj  D  C  (I.  64),  and  the  included  angle  E  A  B  equal  to 
the  included  angle  FD  C  (I.  12),  are  equal.  Adding  D  F  C  to 
the  common  part  ABFD  gives  the  parallelogram  ABC D) 
and  adding  its  equal  AEB  to  the  common  part  ABFD,  gives 
the  rectangle  AEFD;  therefore  the  parallelogram  A  B  C  D  is 
equivalent  to  the  rectangle  AEFD;  but  the  area  of  the  rec- 
tangle z=  A  D  X  D  F  (J);  therefore  the  area  of  the  parallelo- 
gram =zAD  X  D F. 

THEOREM  IV. 

11,    The  area  of  a  triangle  is  equal  to  half  the  product  of  its 
base  and  altitude.  , 

Let  BDhe  the  altitude  of  the  triangle  ABC;  then  the  area 
of  ABC=:iACX  BD. 


34 


PLANE  GEOMETRY. 


Draw  CE  parallel  to  AB,  and  BE 
parallel  to  A  C,  forming  the  parallelogram 
ABEC.  The  triangle  A  B  C  i^  one  half 
the  parallelogram  ABEC  (I.  63) ;  the 
area  of  the  parallelogram  z=z  AC  y^  B  D 
(10) ;  therefore  the  area  of  the  triangle  =  J  ^  (7  X  B  D. 

12.  Cor.  1.  Triangles  are  to  each  other  as  the  products  of 
their  bases  and  altitudes.  For  if  A  and  a  represent  the  alti- 
tudes of  two  triangles  T  and  t^  and  B  and  h  their  bases,  their 
areas  are  J  ^  X  B  and  \a  y^.h;  therefore 

T  -.t^z^Ay  B  \\ayb 
or  (Pn.  21)  T  '.t=zAX  B  '.aXh 

13*  Cor.  2.  Triangles  having  equal  bases  are  as  their  alti- 
tudes ;  those  having  equal  altitudes  as  their  bases.  For  in  the 
proportion  above,  if  ^  =  6,  or  ^  =  a,  the  equals  can  be  can- 
celled from  the  second  ratio  (Pn.  21). 


THEOREM  V. 

14.  The  area  of  a  trapezoid  is  equal  to  half  the  product  of  its 
altitude  and  the  sum  of  its  parallel  sides. 

Let  EE  be  the  altitude  of  the  trape- 
zoid ABCD;  then  the  area  oi  ABCD 
^^EEX  (BC  +  AD). 

Draw  the  diagonal  B  D  ;  it  will  di- 
vide the  trapezoid  into  two  triangles, 
A  B D,  BCD,  having  the  same  alti- 
tude E  E  a^  the  trapezoid. 

By  (11)  the  area  of  BCD=:\  EF  X  BC 

and  the  area  of  A  B  D  z=  ^  E  E  X  A  D 

Therefore  the  area  of  the  trapezoid  =  \  E  E  X  {B  C  -\-  AD). 

15.  Corollary.     As  (I.  66)  the  line  joining  the  middle  points 
of  the  sides  AB  and  CD  of  the  trapezoid  =  ^{BC  -{-  AD), 


BOOK   II.  35 

therefore  the  area  of  a  trapezoid  is  equal  to  the  product  of  its 
altitude  and  the  line  joining  the  middle  points  of  the  sides 
which  are  not  parallel. 

THEOREM  VI. 

16*  A  line  drawn  parallel  to  one  side  of  a  triangle  divides  the 
other  sides  proportionally. 

In  the  triangle  A  B  C  let  D  JS  he  drawn 
parallel  to  B  C ;  then 

Ai:  :£C=AI)  :DB 

Draw  I)  C  and  B  £  ;  the  triangles  A  DB 
and  £J  D  C,  having  the  same  vertex  D  and 
their  bases  in  the  same  straight  line  A  G, 
have  the  same  altitude;  therefore  (13) 

ADE\  EDG  =  AE'.EC 
And  the  triangles  A  D  E  and  D  E  B,  having  the  same  vertex  E 
and  their  bases  in  the  same  straight  line  A  B,  have  the  same 
altitude;  therefore  (13) 

ABE:  DEB  =  AD:  DB 

But  the  triangles  EDO  and  DEB  are  equivalent  (11),  since 
they  have  the  same  base  D  E  and  the  same  altitude,  viz.,  the 
perpendicular  distance  between  the  two  parallels  D  E  and  B  C. 
Therefore  (Pn.  11)       A  E '.  EG  — AD  :  DB 

17.    Corollary,    k^  A  E  :  E  G  ^=A  D  :  D  B 
by  (Pn.  17)      A  E -{- E  G  :  A  Ez=A  D -\- D  B  :  AD 
that  is,  AG:  AEz=AB:  AD 


7 


THEOREM  VII. 

CONVERSE   OF   THEOREM   VI. 

18.    A  line  dividing  two  sides  of  a  triangle  proportionally  is 
parallel  to  the  third  side  of  the  triangle. 

In  the  triangle  ABG  if  D  E  divides  A  B  and  A  G  so  that 
AB  :  AD  =  A  G :  A  E,  then  DEm  parallel  to  B G. 


36 


PLANE   GEOMETRY. 


For  if  D  E  is  not  parallel  to  B  C,  through 
D  draw  D  F  parallel  to  BC;  then  (17) 

AB'.ADz=2AC:AF  A^ 

But  by  hypothesis 

AB.AD  =  AG  :AE 

Now  as  the  first  three  terms  of  these  two 
proportions  are  the  same,  their  fourth  terms  must  be  equal ; 
that  is,  A  F  =  A  E,  the  part  equal  to  the  whole,  which  is  ab- 
surd (Axiom  6) ;  therefore  £>  E  is  parallel  to  B  C. 


19.  Definition.  Similar  Polygons  are  those  which  are  mutu- 
ally equiangular,  and  have  their  homologous  sides,  that  is,  the 
sides  including  the  corresponding  angles,  proportional. 


THEOREM  VIII. 

20.  Two  triangles  mutually  equiangular  are  similar. 
In  the  two  triangles  ABC, 

DEF,  let  the  angle  A  =  D, 
B  =  E,SiiidC  =  F;  then  the 
triangles  are  similar. 

As  the  triangles  are  mutually 
equiangular,  we  have  only  to  prove 
the  homologous  sides  proportional.  Cut  off  A  G  and  A  II  equal 
respectively  to  BE  and  BF,  and  join  Gil;  the  triangle  AG II 
is  equal  to  B  E  F  (I.  40),  and  the  angle  A  G II  =  E ;  but 
E=B;  therefore  A  G II  =  B,  and  Gil  is  parallel  to  ^C 
(I.  18);  and  (17) 

AB:AG==AC:Aff 
or  AB:BE=AC:BF 

In  like  manner  it  may  be  proved  that 

AB:BE  =  BC:EF=:AC:BF 

21.  Cor.     Two  triangles  whose  homologous  sides  are  equally 
inclined  to  each  other  are  similar.     For  if  one  of  the  triangles  is 


BOOK   II. 


37 


turned  through  an  angle  equal  to  the  angle  of  inclination  of 
the  sides,  the  sides  of  the  triangles  become  respectively  parallel ; 
they  are  therefore  mutually  equiangular  (I.  12)  and  similar  (20). 


THEOREM  IX. 

22.    The  altitudes  of  two  similar  triangles  are  proportional  to 

the  homologous  sides. 

Let  BG  and  E  H  he  the  alti- 
tudes of  the  similar  triangles 
ABC  imdi  DEF)   then 

BG'.EH  —  AB'.DE  — 
AC  :DFz=BC  :EF 

For   the    two    right    triangles    ^  G         ^  ^ 

ABG,  DEH  are  mutually  equiangular  (I.  35),  and  similar 
(20) ;  therefore 

BG'.EH=:AB'.DE=AC'.DF=BC:EF 


THEOREM  X. 

23  •  Two  triangles  having  an  angle  of  the  one  equal  to  an  angle 
of  the  other,  and  the  sides  including  these  angles  proportional,  are 
similar. 

In  the  triangles  ABC.DEF 
let  the  angle  A  =  D  and 
AB  '.DE—AC'.DF 

then  the  triangles  ABC  and 
D  E  F  are  similar. 

Cut  off  ^  6^   and  A  II  re- 
spectively equal  to  D  E  and  D  F,  and  join  G  H;  the 
AGH  =  DEF,  and  the  angle  A  G II  =  E  (I.  40). 

By  hypothesis       AB  ':  D E  =  A  C  :  BF 
or  AB  '.AG —  AC  .AH 

that  is,  the  sides  AB,  AC  are  divided  proportionally  by  the 


triangle 


38  PLANE   GEOMETRY. 

line  G  H  -,  therefore  GH  is  parallel  to  B  C  (18),  and  the  angle 
AGH  =  £  (1.17);  hut  the  angle  AGH  =  E;  therefore 
B  =z  E,  and  the  two  triangles  are  mutually  equiangular  and 
therefore  similar  (20). 


THEOREM  XI. 

24  •  In  a  right  triangle  the  perpendicular  drawn  from  the  ver- 
tex of  the  right  angle  to  the  hypothenuse  divides  the  triangle  into 
two  triangles  similar  to  the  whole  triangle  and  to  each  other. 

In  the  right  triangle  A  B  C  if  B  I)  is  B 

drawn  from  the  vertex  B  of  the  right  /^\^ 

angle  perpendicular  to  the  hypothenuse  /  j        ^\^^ 

A  C,  the  two  triangles  A  B  D,  B  G D  2irQ    ^  /_i — ^C 

similar  to  A  B  G  and  to  each  other. 

The  two  right  triangles  A  B  D  and  A  B  G  have  the  acute  an- 
gle A  common  ;  they  are  therefore  mutually  equiangular  (I.  35), 
and  similar  (20).  The  two  right  triangles  A  B  G  and  B  G  D 
have  the  acute  angle  G  common ;  therefore  they  are  mutually 
equiangular  and  similar.  The  two  triangles  A  B  D  and  B  G  D^ 
being  each  similar  to  A  B  G,  are  similar  to  each  other. 

25.  Gor.  1.     Since  A  B  G  and  A  B  I)  sere  similar  triangles 

AG  :AB  =  AB:AJ) 

And  since  A  BG  and  BG D  are  similar 

AG  \GB—GB'.GD 

that  is,  if  in  a  right  triangle  a  perpendicular  is  drawn  from  the 
vertex  of  the  right  angle  to  the  hypothenuse,  either  side  about  the 
right  angle  is  a  mea7i  proportional  between  the  whole  hypothenuse 
and  the  adjacent  segment. 

26.  Gor.  2.     ks  AB  D  and  B  G  D  are  similar  triangles 

AD  ■.DB  =  DB  :DG 


BOOK   II. 


39 


that  is,  in  a  right  triangle  the  perpeiidicidar  from  the  vertex  of 
the  right  angle  to  the  hypothenuse  is  a  mean  proportional  between 
the  segments  of  the  hypothenuse. 


THEOREM  XII. 

27.  The  square  described  on  the  hypothenuse  of  a  right  triangle 
is  equivalent  to  the  sum  of  the  squares  described  upon  the  other  two 
sides. 

Let  ABCheoi  triangle  right- 
angled  at  B ;  then 

Tu'  =  Jb'-{-  bc^ 

On  the  three  sides  construct 
squares,  draw  B  D  perpendicu- 
lar to  A  (7,  and  produce  it  to 
FE\  DC  EL  is  a  rectangle 
whose  area  is  (7) 

C^X  CD=ACy^  CD 
The  area  of  the  square  (9) 
BIKC  —  BU'' 

But  (25)  AG'.BC- 

or  ACXCDzz 

that  is,  the  square  BIKG  is  equivalent  to  the  rectangle  DOEL. 
In  the  same  way  the  square  AG  H  B  can  be  proved  equivalent 
to  the  rectangle  AD L F \  therefore  the  sum  of  the  two  rec- 
tangles, that  is,  the  square  AG  E  F  m  equivalent  to  the  sum  of 
the  squares  BIKG  and  A  G H B ;  or 

AG'  =  AB^  +  bV- 

28.  Gorollary.     Since 

Tg'' =  Tb'-\- BC'' 

bg^=^Tg^  —  Tb' 

BG 


BG 


and 


S/AG'' 


AB^ 


PLANE   GEOMETRY. 


THEOREM  XIII. 

29i    Similar  triangles  are  to  each  other  as  the  squares  of  their 
homologous  sides. 

Let  ABC  SLiid  DBF  be  two  ^ 

similar  triangles ;  then 
ABC  :DBF=AC^  :  ITF^ 

Draw  B  G  and  FII  perpendic- 
ular respectively  to  A  C  and  I)  F ; 
then  (22)  ^ 

BG.EH 
this  multiplied  by  the  proportion 

I  AC  '.IDF=AC   :DF 
gives       \ACXBG:IDFXFH=AC'':DF^ 
hwt  I  AC  X  BG  m  the  area  of  ^  ^  C',  and  ^  D  F  X  E  H  i% 

the  area  of  i>  ^  i^  (11) ;  therefore        

ABC  :J)FFz=zTC^ -.WF.^ 


THEOREM  XIY. 

30*  Similar  polygons  can  he  divided  into  the  same  number  of 
similar  triangles. 

Let  A  B  C  J)  E  F  and 
GHIKLM  be  similar  poly- 
gons ;  they  can  be  divided 
into  the  same  number  of  sim- 
ilar triangles.  F  E 

From  the  homologous  vertices  A  and  G  draw  the  diagonals 
ACj  A  D,  A  E,  G  Ij  G  K,  and  G  L ;  these  diagonals  divide  the 
polygons  as  required.  For,  as  the  polygons  are  similar,  the  an- 
gle B  =  H,  and^^  :  GH=.BC  .HI-,  therefore  the  trian- 
gles ABC  and  G  H I  ixre  similar  (23).  As  the  triangles  ABC 
and  GHI  are  similar,  the  angle  BC  A  =  H I G ',  but  the 
whole  angle  BCD=:  HIK;  therefore  the  angle  ACE==  GIK; 
and  as  the  trian^^les  ABC  and  GHI  are  similar 


BOOK  II.  41 

BC  '.HI=AC  '.GI 
But  BC  :HI=zCD  :IK 

Therefore  AC  \GI=CD  .IK 

and  AC  B  and  G I K  2xq  similar  (23).  In  like  manner  it  can 
be  proved  that  the  other  triangles  are  similar  each  to  each. 

THEOREM  XV. 

31*  The  perimeters  of  similar  polygons  are  to  each  other  as  the 
homologous  sides  ;  and  the  polygons  as  the  squares  of  the  howMo- 
gous  sides. 

Let     ABC  D  E  F    and          ^^^-tt^x                 ^      I 
GHIKLM  hQ  two  similar        /C^^^^^^^^'^X         /^^\ 
polygons.  A^".:i[ 7      ^  ^v---- -  ^ 

1st.    Their  perimeters  are        \       ""--,.    /  \- — -1:1;^ 

to  each  other  vl^  AB  :  GH  F  E    * 

For  as  the  polygons  are  similar 

AB  :  GH=BC  '.HI=  CD  .IK,  (fee. 
Therefore  (Pn.  23) 

AB^  BC-\-CD,kQ.'.  GH-^HI-\-IK,kc.  =  AB  :GH 
that  is,  the  perimeters  oi  ABC  D  EF  and  GHIKLM  are  as 
AB  :  GH. 

2d.        A  B  C  D  E  F  :  G  H I K  L  M  =zAB^  :  gTi'' 

From  the  homologous  vertices  A  and  G  draw  the  diagonals 
AC,  AD,  AE,GI,GK,2indiGL',  the  polygons  will  be  divided 
into  the  same  number  of  similar  triangles  (30) ;  therefore  (29) 

ABC  :  GUI=TC^  :~I^ 
and  ACD  '.GIK=irZ''.G~f 

Therefore  AB  C  \  G  H I  =  AC  D  -.  GIK 

In  like  manner  AC  D  :  G I K  =  AD  E  -  G  K  L 

and  ADE'.GKL  =  AEF'.GLM 

Hence  (Pn.  23) 
ABC-\-ACD-\-  ADE  +  AEF  :  GHI  +  GIK  +  GKL  + 

GLM=ABC.GHI 
But  ABC  :GHIz=TB^:GH'' 


42 


PLANE   GEOMETRY. 


Therefore  the  sums  of  the  triangles,  that  is,  the  polygons 
themselves,  are  to  each  other  as  the  squares  of  the  homologous 
sides. 

32.  Definition.  A  Regular  Polygon  is  one  that  is  both  equi- 
angular and  equilateral. 

THEOREM  XVI. 
33*    Regular  polygons  of  the  same  number  of  sides  are  similar. 

Let  A  BCD IJF  ^nd  y f  ^ 

GHIKLM  be  two  reg- 
ular polygons  of  the 
same  number  of  sides  ; 
they  are  similar. 

They    are    mutually  p  e 

equiangular;  for  the  sum  of  their  angles  is  the  same  (I.  67); 
and  each  angle  is  equal  to  this  sum  divided  by  the  number  of 
angles  which  is  the  same. 

The  homologous  sides  are  proportional ;  for  as  the  polygons 
are  regular,  AB  =  £C=CD,  &c.,  and  GH=HI  =  IK, 
&c.,  therefore  ^^  :  GH  =  BC  :  HI=CD  :  IK,  &c. 


THEOREM  XVII. 


34.    There  is  a  point  in  a  regular  polygon  equidistant  from  its 
vertices,  and  also  equidistant  from  its  sides. 

Let  ABCDlEF  be  a  regular  polygon. 
Bisect  the  angles  A  and  B  hy  A  0  and 
£  0.  As  the  whole  angles  A  and  B  are 
each  less  than  two  right  angles,  the  sum 
of  0  A  B  and  ABO  is  less  than  two 
rig^t  angles ;  therefore  A  0  and  B  0  can- 
not be  parallel  (I.   17),  but  will   meet. 


9f7: 


BOOK  II.  '  43 

Suppose  them  to  n]@>an  the  point  0  \  then  0  is  equidistant 
from'  the  vertices  A,  B,  C,  D,  E,  F,  and 
also  from  the  sides  A  B,  B  C,  CD,  &i 

Draw  OC,  OB,  OE,  OF.  OA  =  OB 
(I.  45).  As  0  B  bisects  the  whole  angle 
B,  the  angle  0  B  A  =  0  B  C ;  therefore 
the  triangle  A  B  0  =  0  B  C  (I.  40),  and 
OC  =OAz=  OB.  In  like  manner  it 
can  be  proved  that  OD=OE=OF=zOA 
equidistant  from  the  vertices  of  the  polygon. 

As  the  triangles  0  A  B,  0  B  C,  0  C  D,  &c.  are  equal,  their 
altitudes  are  equal,  that  is,  the  bases  are  equidistant  from  the 
vertex  0. 

35*  Scholium.  0  is  called  the  centre,  and  the  perpendicular 
0  G  the  apothem  of  the  polygon. 

36*  Corollary.  In  regular  polygons  of  the  same  number  of 
sides,  the  apothems  are  as  the  homologous  sides ;  therefore  the 
perimeters  of  regular  polygons  of  the  same  number  of  sides  are  as 
their  apothems  ;  and  the  polygons  as  the  squares  of  their  apothems. 


THEOREM  XVIII. 

37.  The  area  of  a  regular  polygon  is  equal  to  half  the  product 
of  its  perimeter  and  apothem. 

For,  if  a  regular  polygon  is  divided  into  triangles  by  lines 
drawn  from  the  centre  to  the  several  vertices,  the  area  of  each 
triangle  is  equal  to  half  the  product  of  its  base  and  the  apo- 
them of  the  polygon  (11);  therefore  the  area  of  the  polygon 
is  equal  to  half  the  product  of  the  sum  of  the  bases,  that  is, 
to  half  the  product  of  its  perimeter  and  its  apothem. 


44  PLANE  GEOMETEY. 

PRACTICAL   QUESTIONS. 

1.  What  is  the  perimeter  and  the  area  of  a  rectangle  25  by  35  inches  ? 

2.  What  is  the  area  of  a  parallelogram  whose  base  is  20  feet  and  altitude 
12  feet  ? 

3.  What  is  the  area  of  a  triangle  whose  base  is  14  feet  and  altitude  8 
feet? 

4.  What  is  the  square  surface  of  a  boq,rd  15  feet  long,  and  16  inches  wide 
at  one  end  and  9  inches  at  the  other  ?    What  kind  of  a  figure  is  it  ? 

6.  What  integral  numbers  will  express  the  sides  and  hypothenuse  of  a 
right  triangle  ? 

6.  How  far  from  a  tower  40  feet  high  must  the  foot  of  a  ladder  50  feet 
long  be  placed  that  it  may  exactly  reach  the  top  of  the  tower  ? 

7.  The  foot  of  a  ladder  67  feet  long  stands  40  feet  from  a  wall  ;  how 
much  nearer  the  wall  must  the  foot  be  placed  that  the  ladder  may  reach  10 
feet  higher  ? 

8.  If  a  ladder  108  feet  long,  with  its  foot  in  the  street,  will  reach  on  one 
side  to  a  window  75  feet  high,  and  on  the  other  to  a  window  45  feet  high, 
how  wide  is  the  street  ? 

9.  A  has  an  acre  of  land  one  of  whose  sides  is  20  rods  in  length  ;  B  has 
a  piece  of  land  of  exactly  similar  form  containing  9  acres.  What  is  the 
length  of  the  corresponding  side  of  B's  ? 

10.  What  is  the  distance  on  the  floor  from  one  corner  to  the  opposite 
comer  of  a  rectangular  room  16  by  24  feet  ? 

11.  If  the  height  of  the  above  room  is  10  feet,  what  is  the  distance 
from  the  lower  corner  to  the  opposite  upper  comer  ? 

12.  Find  the  length  of  the  longest  straight  rod  that  can  be  put  into  a 
box  whose  inner  dimensions  are  12,  4,  and  3. 

13.  What  is  the  altitude  of  an  equilateral  triangle  whose  side  is  12  feet  ? 

14.  If  the  bases  of  two  similar  triangles  are  respectively  100  and  10  feet, 
how  many  triangles  equal  to  the  second  are  equivalent  to  the  first  ? 

15.  How  many  times  as  much  paint  will  it  take  to  cover  a  church  whose 
steeple  is  120  feet  in  height  as  to  cover  an  exact  model  of  the  church  whose 
steeple  is  10  feet  in  height  ? 

16.  What  is  the  area  of  a  right-angled  triangle  whose  hypothenuse  is 
125  feet  and  one  of  the  sides  75  feet  ? 


BOOK  II. 


45 


EXERCISES. 


The  following  Theorems,  depending  for  their,  demonstration  upon 
those  already  demonstrated,  are  introduced  as  exercises  for  the  pupil. 
In  some  of  them  references  are  made  to  the  propositions  upon  which 
the  demonstration  depends.  They  are  not  connected  with  the  prop- 
ositions in  the  following  books,  and  can  be  omitted  if  thought  best. 

D  C   E 

38.    The  square  on  the  sum  J.  C  of  two  straight 

fines  A  B,  B  C  is  equivalent  to  the  squares  on  A  B 
(tnd  B  C,  together  with  twice  the  rectangle 
AB.BC. 

Or,  algebraically,  ii^  a  =  AB,  and  b  =  B  C, 

{a-{-bf  =  a?-\-2ab-\-b^  A  B~  C 

39*  Corollary.  The  square  on  a  line  is  four  times  the  square  on 
naif  of  the  line. 


ab 

!^ 

F 

/ 

H 

a? 

ab 

40»  The  square  on  the  diJBference  J.  (7  of 
two  straight  lines  AB^  B  C  is  equivalent  to  the 
squares  on  AB  and  BC.  diminished  by  twice 
the  rectangle  AB.BC. 

Or,  algebraically,  if  a=  AB,  and  b  =.  BC, 
{a  —  by  =  a*  —  2  ab  -\-  h" 

41  *  The  rectangle  contained  by  the  sum  and 
difference  of  two  lines  A  B,  B  C  is  equivalent  to 
the  difference  of  their  squares. 

Or,  algebraically,  if  a  ^  ^  ^  and  b  =  BC 
(a  -I-  6)  (a  --  6)  =  a«  —  b' 

Produce  ABso  that  BD  =  BC. 


K  D 


G   E 


L 

F            1 

H 

A 
E 

G    B 

0   B 

T 

L 

K            I 

H 

C   B    D 


42*    Parallelograms  are  to  each  other  as  the  products  of  their  bases 
and  altitudes.     (10.) 

43 »    Parallelograms  having  equal  bases  are  to  each  other  as  their 
altitudes;  those  having  equal  altitudes  are  as  their  bases. 

44.  Where  must  a  line  from  the  vertex  be  drawn  to  bisect  a  tri- 
angle?    (13.) 

45.  Two  or  more  lines  parallel  to  the  base  of  a  triangle  divide  the 
other  sides,  or  the  other  sides  produced,  proportionally. 


46  PLANE   GEOMETRY. 

46*  Lines  joining  the  middle  points  of  the  adjacent  sides  of  a 
quadrilateral  form  a  parallelogram ;  and  the  perimeter  of  this* paral- 
lelogram is  equal  to  the  sum  of  the  diagonals  of  the  quadrilateral. 

Draw  the  diagonals.     (18.) 

47«  Lines  drawn  from  the  vertex  of  a  triangle  divide  the  opposite 
side  and  a  parallel  to  it  proportionally. 

48«    State  and  prove  the  converse  of  47. 

49t  A  B  CD  is  a  parallelogram ;  E  and  F  the  middle  points  of 
A  B  and  CD.     BF  and  Z>^  trisect  the  diagonal  A  G. 

50*  If  two  triangles  have  two  sides  of  the  one  equal  respectively 
to  two  sides  of  the  other,  and  the  included  angles  supplementary,  the 
triangles  are  equivalent. 

51*  The  diagonals  divide  a  parallelogram  into  four  equivalent  tri- 
angles.    Two  triangles  standing  on  opposite  sides  are  equal. 

52*  If  the  middle  points  of  the  sides  of  a  triangle  are  joined,  the 
area  of  the  triangle  thus  formed  is  one  fourth  the  area  of  the  original 
triangle. 

53*  Every  line  passing  through  the  intersection  of  the  diagonals 
of  a  parallelogram  bisects  the  parallelogram. 

54*  If  a  point  within  a  parallelogram  is  joined  to  the  vertices,  the 
two  triangles  formed  by  the  joining  lines  and  two  opposite  sides  are 
together  equivalent  to  half  the  parallelogram. 

Through  the  point  draw  lines  parallel  to  the  sides  of  the  parallelo- 
gram. 

^5t  State  and  prove  the  proposition  if  the  point  named  in  54  is 
without  the  parallelogram. 

56*  The  area  of  a  trapezoid  is  equal  to  twice  the  area  of  the  tri- 
angle formed  by  joining  the  extremities  of  one  non-parallel  side  to  the 
middle  point  of  the  other. 

57*  Two  triangles  are  similar  if  two  angles  of  the  one  are  equal 
respectively  to  two  angles  of  the  other. 

58*  Two  triangles  are  similar  if  their  homologous  sides  are  pro- 
portional. 


BOOK  IL 


47 


59«  Definition.  When  a  point  is  taken  on  a  given  line,  or  a 
given  line  produced,  the  distances  of  the  point  from  the  extremities 
of  the  hne  are  called  the  segments.  If  the  point  is  within  the  given 
line,  the  sum  of  the  segments,  if  in  the  line  produced,  the  difference 
of  the  segments,  is  equal  to  the  line, 

60i  The  line  bisecting  any  angle,  interior  or  exterior,  of  a  triangle, 
divides  the  opposite  side  into  segments;  which  are  proportional  to  the 
adjacent  sides. 

Let  B  be  the  bisected  angle  of  a  triangle  ABC.  Through  G 
draw  a  line  parallel  to  the  bisecting  line  and  meeting  A  B.  If  the 
interior  angle  at  B  is  bisected,  A  B  must  be  produced ;  if  the  exterior 
angle,  A  C.  In  the  latter  case,  if  E  is  the  point  where  the  bisecting 
line  meets  A  C produced,  the  segments  of  the  base  (59)  are  ^^  ami 
CE.     (1.17.)     (1.45.)     (16.) 

61.  Two  triangles  having  an  angle  of  the 
one  equal  to  an  angle  in  the  other  are  to  each 
other  as  the  rectangles  of  the  sides  containing 
the  equal  angles  ;  or 

ABC:ADE=ABXAC:ADX  AE 
Draw  BE.     (13.)     (Pn.  24.)     (Pn.  21.) 

62.  Prove  Theorem  XII.,  first 
drawing  G  Cand  BF;  then  prov- 
ing the  triangles  AGO  and  A  BF 
equal. 

Turn  the  triangle  A  BF  on  the 
point  A  in  its  own  plane  till  A  B 
coincides  with  A  G ;  where  will 
7^  be?     (7,11.) 

63.  Prove  that  if  G  H,  KI, 
and  L  B,  in  the  figure  above,  are 
produced,  they  will  meet  in  the 
same  point. 

64.  Prove  Theorem  XII.,  first  producing  FA  to  GIT,  and  pro- 
ducing Gff,  KI,  and  i^^  till  they  meet. 

65.  Prove  Theorem  XII.,  first  constructing  the  squares  on  oppo- 
site sides  of  ^  i?  and  B  C  from  that  on  which  they  are  drawn  in  the 
figure  in  Art.  62;  moving  the  square  A  GHB  on  AB,  i\.  distance 


48  PLANE  GEOMETRY. 

equal  to  i?  C  in  the  direction  BA ;  then  proving  that  these  squares 
are  divided  into  parts  that  can  be  made  to  coincide  with  the  parts  of 
the  square  on  A  C. 


B 


66.    If  A  is  an  acute  angle  of  the  triangle  ABC^ 
and  BD  is  the  perpendicular  from  i?  to  J.  (7,  then 
BC^=zAB^-\-AC^  —  2ACX  AD 

67«    If  A  is  an  obtuse  angle  of  the  triangle     B 
ABC,  and  BD  is  the  perpendicular  from  B 


BC'  =  AB^-{-A0'  +  2A0X  AD  ^ 

68.  Show  that  if  the  angle  A  becomes  a  right  angle,  both  66 
and  67  reduce  to  the  same  as  27 ;  and  if  0  becomes  a  right  angle, 
both  reduce  to  the  same  as  the  second  equation  in  28. 

69»  If  a  line  is  drawn  from  the  vertex  of  any  angle  of  a  triangle 
to  the  middle  of  the  opposite  side,  the  sum  of  the  squares  of  the  other 
two  sides  is  equivalent  to  twice  the  square  of  the  bisecting  line  to- 
gether with  twice  the  square  of  a  segment  of  the  bisected  side. 

Draw  a  perpendicular  from  the  same  vertex  to  the  opposite  side. 
(66,  67.) 

70»  The  sum  of  the  squares  of  the  four  sides  of  a  parallelogram  is 
equivalent  to  the  sum  of  the  squares  of  thef  diagonals.     (69.)     (39.) 

71.  In  the  figure  in  Art.  62  draw  HI,  KB,  FG.  The  triangle 
HIB  is  equal,  and  the  triangles  CKF,  GAF are  equivalent  to  ABC. 

72.  The  squares  of  the  sides  of  a  right  triangle  are  as  the  seg- 
ments of  the  hypothenuse  made  by  a  perpendicular  from  the  vertex 
of  the  right  angle. 

73.  The  square  of  the  hypothenuse  is  to  the  square  of  either  side 
as  the  hypothenuse  is  to  the  segment  adjacent  to  this  side  made  by  a 
perpendicular  from  the  vertex  of  the  right  angle. 

74.  The  side  of  a  square  is  to  its  diagonal  as  1 :  V^2;  or  the  square 
described  on  the  diagonal  of  a  square  is  double  the  square  itself. 

75.  (Converse  of  30.)  Two  polygons  composed  of  the  same  num- 
ber of  similar  triangles  similarly  situated  are  similar. 


BOOK    III. 


THE   CIECLE. 
DEFINITIONS. 

1.  A  Circle  is  a  plane  figure  bounded  by  a  curved  line  called 

the  circumference,  every  point  of  which  is  equally  distant  from  a 

point  within  called  the  centre  ;  ^^  A  B  D  E, 

a 

2.  The  Radius  of  a  circle  is  a  line 

drawn  from  the  centre  to  the  circum- 
ference ;  as  C  J). 

3t  The  Diameter  of  a  circle  is  a  line 
drawn  through  the  centre  and  termi- 
nating at  both  ends  in  the  circumfer- 
ence ;  as  ^  i>. 

4«    Corollary.     The  radii  of  a  cir- 
cle, or  of  equal  circles,  are  equal ;  also  the  diameters  are  equal, 
and  each  is  equal  to  double  the  radius. 

5*   An  Arc  is  any  part  of  the  circumference ;  as  A  F  B. 

6*  A  Chord  is  the  straight  line  joining  the  ends  of  an  arc ; 
as  A  B. 

7..  A  Segment  of  a  circle  is  the  part  of  the  circle  cut  off  by 
a  chord  ;  as  the  space  included  by  the  krc  AF B  and  the  chord 
AB. 

8.  A  Sector  is  the  part  of  a  circle  included  by  two  radii  and 
the  intercepted  arc ;  as  the  space  BCD. 

9.  A  Tangent  (in  geometry)  is  a  line  which  touches,  but 
does  not,  though  produced,  cut  the  circumference ;  as  G  B. 


50 


PLANE   GEOMETRY. 


A  tangent  is  often  considered  as  terminating  at  one  end  at 
the  point  of  contact,  at  the  other  where  it  meets  another  tan- 
gent or  a  secant. 

10.  A  Secant  (in  geometry)  is  a  line  lying  partly  within  and 
partly  without  a  circle ;  ^^  G  E. 

A  secant  is  generally  considered  as  terminating  at  one  end 
where  it  meets  the  concave  circumference,  and  at  the  other 
where  it  meets  another  secant  or  a  tangent. 


THEOEEM  I. 

11,  In  the  same  circle,  or  equal  circles,  equal  angles  at  the  cen- 
tre are  subtended  by  equal  arcs ;  and,  conversely,  equal  arcs  sub- 
tend equal  angles  at  the  centre. 

Let  B  and  E  be  equal 
angles  at  the  centres  of 
the  two  equal  circles 
ACG  and  DFH]  then 
the  arcs  AC  and  DF 
are  equal. 

Place  the  angle  ^  on  q 

the  angle  E ;  as  they  are  equal  they  will  coincide  ;  and  as  B  A 
and  B  C  are  equal  to  E  D  and  E  F,  the  point  A  will  coincide 
with  D,  and  the  point  C  with  F ;  and  the  arc  A  G  will  coincide 
with  D  F,  otherwise  there  would  be  points  in  the  one  or  the 
other  arc  imequally  distant  from  the  centre. 

Conversely.  If  the  arcs  A  C  and  D  F  are  equal,  the  angles 
B  and  E  are  equal. 

For,  if  the  radius  ^  ^  is  placed  on  the  radius  D  E  with  the 
point  B  on  E,  the  point  A  will  fall  on  i>,  ^%  A  B  =^  D  E  -,  and 
the  arc  A  C  will  coincide  with  D  F,  otherwise  there  would  be 
points  in  the  one  or  the  other  arc  unequally  distant  from  the 
centre  ;  and  as  the  wcq  A  C  =.  D  F,  the  point  C  will  fall  on  F ; 
therefore  B  G  will  coincide  with  E  F,  and  the  angle  B  be  equal 
toi;. 


BOOK  m. 


51 


THEOREM  II. 

12o  in  the  same  or  equal  circles,  equal  chords  subtend  equal 
arcs  ;  and,  conversely,  equal  arcs  are  subtended  by  equal  chords. 

Let  ABC  and 
D  E  F  he  two 
equal  circles ;  if 
the  arcs  AB  and 
D  E  are  equal, 
the  chords  A  B 
and  D  E  are 
equal ;  and  conversely,  if  the  chords  A  B  and  D  E  are  equal, 
the  arcs  A  B  and  D  E  are  equal. 

For,  if  the  centre  of  the  circle  A  BC  h  placed  on  the  centre 
o{  D  E F  with  the  point  A  of  the  circumference  on  the  point  I), 
if  the  arcs  or  the  chords  are  equal,  B  will  fall  on  E ;  and  in 
either  case  the  chords  and  arcs  will  coincide,  otherwise  there 
would  be  points  in  the  one  or  the  other  circumference  unequally 
distant  from  the  centre. 


THEOREM  III. 


13i    Angles  at  the  centre  vary  as  their  corresponding  arcs, 

IjQtACD,  DCE,  ECFhe  equal  an- 
gles at  the  centre  C ;  then  the  arcs  A  £>, 
D  E,  E  F  &re  equal  (11);  then  the  an- 
gle ACE  is  double  the  angle  A  C  D, 
and  the  arc  A  E  double  the  arc  A  D ; 
and  the  angle  A  C  F  is  three  times  the 
angle  AC  B,  and  the  arc  A  F  three 
times  the  arc  AB\  and  if  the  angle  A  C  G  \9,  m  times  the 
angle  A  CD,  the  arc  A  G  \%  m  times  the  arc  AD',  that  is,  the 
angle  varies  as  the  arc,  or  the  arc  as  the  angle. 


52  PLANE  GEOMETRY. 

14«  Cor.  1.  As  angles  at  the  centre  vary  as  their  arcs,  or 
arcs  as  their  corresponding  angles,  either  of  these  quantities 
may  be  assumed  as  the  measure  of  the  other.  The  measure  of 
an  angle  is,  then,  the  arc  included  between  its  sides  arid  described 
from  its  vertex  as  a  centre. 

\5%  Cor.  2.  As  the  sum  of  all  the  angles  about  the  point 
C  is  equal  to  four  right  angles  (I.  9),  one  right  angle,  H  C  A,  is 
measured  by  one  quarter  of  the  circumference,  or  by  a  quadrant. 


THEOREM  IV. 

16.  The  radiics  perpendicular  to  a  chord  bisects  the  chord  and 
the  arc  subtended  by  the  chord. 

Let  C  E  hQ  a*  radius  perpendicular  to 
the  chord  ^  ^ ;  it  bisects  the  chord  A  B, 
and  also  the  arc  A  E  B. 

Draw  the  radii  C  A  and  C  B  and  the 
chords  A  E  and  E  B.     As  equal  oblique 
lines  are  equally  distant  from  the  perpen- 
dicular, AD  —  DB{1.^2);  and  as  E  is  E 
a  point  in  the  perpendicular  to  the  middle  of  A  B,  it  is  equally 
distant  from  A  and  B  (I.  53) ;  therefore  the  chords  and  hence 
(12)  the  arcs  A  E,  E B  are  equal. 

17.  Corollary.  The  perpendicular  to  the  middle  of  a  chord 
passes  through  the  centre  of  the  circle,  and  of  the  arc ;  and  the 
radius  drawn  to  the  centre  of  an  arc  bisects  its  chord  perpendic- 
ularly. 

DEFINITIONS. 

18.  An  Inscribed  Angle  is  one  whose  vertex  is  in  the  circum- 
ference and  whose  sides  are  chords  ;  ^^ABC'm  the  outer  circle 

19.  An  Inscribed  Polygon  is  one  whose  sides  are  chords. 


BOOK  III. 


53 


Thus  ABGDEF  is  inscribed  in  the  outer  circle, 
the  circle   is  said  to  be  circumscribed 
about  the  polygon. 

20.    A    Circumscribed    Polygon    is 

one  whose  sides  are  tangents.  Thus 
ABC D E F \%  circumscribed  about  the 
inner  circle.  In  this  case  the  circle  is 
said  to  be  inscribed  in  the  polygon. 


THEOREM  V. 

21 1    An  inscribed  angle  is  measured  by  half  the  arc  included  by 
its  sides. 


1st.  When  one  of  the  sides  B  D  is  a. 
diameter ;  then  the  angle  B  is  measured 
by  half  the  arc  A  D.  Draw  the  radius 
C A,  and  the  triangle  AC B  is  isosceles, 
C  A  and  C  B  being  radii ;  therefore  the 
angle  A  =  B  (I.  42).  But  the  exterior 
angle  AC D  is  equal  to  the  sum  of  the 
two  angles  A  and  B  (I.  39) ;  therefore  the 
angle  B  is  equal  to  half  the  angle  AC D  -,  the  angle  AC D  is 
measured  by  the  arc  J  i>  (14) ;  therefore  the  angle  B  is  meas- 
ured by  half  the  arc  A  D. 


2d.  When  the  centre  is  within  the 
angle,  draw  the  diameter  B  C.  By  the 
preceding  part  of  the  proposition  the  an- 
gle ^  5  (7  is  measured  by  half  the  arc 
^(7,  and  (7^  Z)  by  half  CD;  therefore 
ABC  -\-CBD,ox  ABD.is  measured 
by  half  AC -\- CD,  ox  half  the  arc  A  D. 


54 


PLANE   GEOMETRY. 


3d.  When  the  centre  is  without  the 
angle,  draw  the  diameter  B  C.  By  the 
first  part  of  the  proposition  the  an- 
gle ABC  is  measured  by  half  the  arc 
AC,  wcidi  BBC  by  half  DC;  therefore 
ABC  —  DBC,  or  A  B D,  is  measured 
XjhtliAC  —  DC,  or  half  the  arc  A  D. 

22.  Cor,  1.  All  the  angles  ABC, 
ADC,  inscribed  in  the  same  segment  are 
equal;  for  each  is  measured  by  half  the 
arc  A  EC 

23  •  Cor.  2.  Every  angle  inscribed  in 
a  semicircle  is  a  right  angle;  for  it  is 
measured  by  half  a  semi-circumference, 
or  by  a  quadrant  (15). 


THEOREM  yi. 
24.    Every  eqvMateral  polygon  inscribed  in  a  circle  is  regular. 

Let   ABC  DEE  be   an  equilateral  B  ^ .0 

polygon  inscribed  in  a  circle ;  it  is  also 
equiangular  and  therefore  regular. 

For  the  chords  ^  ^,  BC,  CD,  &c. 
being  equal,  the  arcs  A  B,  BC,  CD, 
&c.  are  ^qual  (12);  therefore  the  arc 
AB  -\-  the  arc  B  C  will  be  equal  to  the 
arc  BC  -{-  the  arc  C D,  &c, ;  that  is,  the  angles  B,  C,  &c.  aTe 
in  equal  segments  ;  therefore  they  are  equal  (22),  and  the  poly- 
gon is  equiangular  and  regular. 


THEOREM  VII. 

25*    An  infinitely  small  chord  coincides  with  its  arc. 

Let  ^  ^  be  an  infinitely  small  chord ;  it  coincides  with  the 
arc  A  D  B. 


BOOK  III. 

"Draw  the  diameter  CD  perpendicu- 
lar to  the  chord  A  B ;  and  draw  A  C 
and  AD ;  CAD  is  a  right-angled  tri- 
angle (23) ;  therefore  (II.  26) 

CE  '.AEz=AE  :ED 
that  is,  E  D  i^  the  same  part  of  AE  that 
AE  h  of  C  E.  But  ^  ^  is  half  the  infinitely  small  chord 
A  B  (16),  and  ^  ^  is  infinitely  small  in  comparison  with  C  E  -, 
therefore  EDh  infinitely  small  in  comparison  with  A  E^  that 
is,  the  point  E  is  on  D,  and  the  chord  A  B  coincides  with  the 
axcADB. 

THEOREM  VIII. 

26*  A  circle  is  a  regular  ^polygon  of  an  infinite  number  of 
sides. 

If  the  circumference  of  a  circle  is  divided  into  equal  arcs, 
each  infinitely  small,  the  infinitely  small  chords  of  these  arcs 
would  form  a  regular  polygon  (24)  of  an  infinite  number  of 
sides ;  and  as  each  chord  would  coincide  with  its  arc  (25),  the 
polygon  would  be  the  circle  itself 

27.  Scholium.  It  might  be  supposed  that  although  the  dif- 
ference between  each  chord  and  its  arc  is  infinitesimal,  yet  as 
there  is  an  infinite  number  of  these  differences  their  sum  would 
not  be  infinitesimal  and  ought  not  to  be  neglected ;  that  is, 
that  the  perimeter  of  the  polygon  and  the  circumference  of 
the  circle  differed  by  a  finite  quantity.  But  each  chord  is  in- 
finitely small  compared  with  the  diameter  of  the  circle,  or  is 
equal  to  7-7;  and  the  difference  between  each  chord  and  its 
arc  is  infinitely  smaller  than  the  chord  itself,  or  is  equal  to 

— - ;  and  an  infinite  number  of  these  differences  is  equal 

Inf.  X  Inf 

to  -p-^ ,  X  Dif.  =  T-7  ;  that  is,  the  difference  between  the 

Inf.  X  Inf.  *^  Inf. 

perimeter  of  the  polygon  and  the  circumference  of  the  circle  is 

infinitesimal. 


56  PLANE   GEOMETRY. 

THEOREM  IX. 

28  •  Circumferences  of  circles  are  to  each  other  as  their  radii, 
or  as  their  diameters  ;  and  the  circles  themselves  as  the  squares  of 
their  radii,  or  the  squares  of  their  diameters. 

For  circles  are  regular  polygons  of  an  infinite  number  of  sides 
(26) ;  and  if  the  circumferences  of  circles  are  divided  into  the 
same  infinite  number  of  arcs,  the  polygons  formed  by  their 
chords,  that  is,  the  circles  themselves,  are  regular  polygons  of 
the  same  number  of  sides,  and  are  therefore  similar  (II.  33) ; 
and  the  apothems  of  the  polygons  are  the  radii  of  the  circles ; 
therefore  the  circumferences  of  the  circles  are  as  their  radii 
(II.  36),  or  as  twice  their  radii,  that  is,  as  their  diameters ;  and 
the  circles  themselves  as  the  squares  of  their  radii,  or  the 
squares  of  their  diameters. 

29.  Cor.  1.  If  (7  and  c  denote  the  circumferences,  i?  and  r 
the  corresponding  radii,  and  D  and  d  the  corresponding  diame- 
ters, we  have 

C  :  c  —R  :rz=D  :  d 

or  C  '.  R  =  c  :  r 

and  C  '.  D  ^=  c  \  d 

That  is,  the  ratio  of  the  circumference  of  every  circle  to  its  ra- 
dius or  to  its  diameter  is  the  same,  that  is,  is  constant.  The 
constant  ratio  of  the  circumference  to  its  diameter  is  denoted 
by  TT  (the  Greek  letter  p). 

C 


Cor.  2.  ^ 


C  =  7rJD  =  27rR 


THEOREM  X. 


31 1    The  area  of  a  circle  is  equal  to  half  the  product  of  its  cir- 
c^imference  and  its  radius. 

The  area  of  a  regular  polygon  is  half  the  product  of  its  perim- 
eter and  its  apothem  (II.  37) ;   a  circle  is  a  regular  polygon 


BOOK  III.  57 

of  an  infinite  number  of  sides  (26) ;  the  circumference  of  the 
circle  is  the  perimeter  of  the  polygon,  and  its  radius  is  the 
apothem ;  therefore  the  area  of  a  circle  is  half  the  product  of 
its  circumference  and  its  radius. 

32.    Corollary.     If  (7  =  the  circumference,  D  =  the  diame- 
ter, B  =  the  radius,  and  A  =.  the  area  of  a  circle,  we  have 

A  =  \G  X R 

But  (30)  (7=:27r^  =  7rZ> 

Therefore  A  =  lX^'ivRxK  =  nR'' 

or  A^z^^I)  X^  =  lirD^ 


M^ 


THEOREM  XI.  C  'TT^f^^ 

33*  The  side  of  a  regular  hexagon  inscribed  in  a  circle  is  equal 
to  the  radius  of  the  circle. 

In  the  circle  whose  centre  is  C  draw  the 
chord  A  B  equal  to  the  radius ;  ^  ^  is  the 
side  of  a  regular  hexagon  inscribed  in  a 
circle. 

Draw  the  radii  CA  and  GB;  CAB'm 
an  equilateral,  and  therefore  an  equiangu- 
lar triangle  ;  hence  the  angle  C  is  equal  to 
one  third  of  two  right  angles,  or  one  sixth  of  four  right  angles  : 
that  is,  the  arc  ^  ^  is  one  sixth  of  the  whole  circumference, 
or  the  chord  A  B  the  side  of  a  regular  hexagon  inscribed  in  the 
circle  (12  and  24). 

34.  Corollary.  The  chord  of  half  the  arc  A  B  would  be  the 
side  of  a  regular  dodecagon ;  the  chord  of  one  quarter  of  the 
arc  A  B,  the  side  of  a  regular  polygon  of  twenty-four  sides  j  and 
so  on. 

3* 


68 


PLANE  GEOMETRY. 


PROPOSITION   XII. 
PROBLEM. 

35*    The  chord  of  an  arc  given  to  find  the  chord  of  half  the  arc. 

Let  AB  he  the  given  chord,  A  D  the 
chord  of  half  the  arc  ADB,  and  R  denote 
the  radius. 

Draw  the  diameter  DF,  the  radius  AC, 
and  the  chord  A  F.  The  triangle  ADF 
is  right  angled  at  A  (23) ;  then  (II.  25) 

DF:AI)  =  AD  :DF 
or  AD''  =  DFXI>E='1I{XDE 

and  AD  =  sJYYy^WE 

Now  DE  =  DG  —GE  =  R  —  GE 

and  (11.  28)  GE  =  \l A  G^  —  A  E'  z=z  sjE^  —  A  FA 

therefore  DE  =  R  —  sjR^  —  A  E^ 

Substituting  this  value  of  D  E  in 

AD  =  sI'iYyODE 

we  have 

AD  =  Sj2  R^  —  2R  s/R'  —  A  E^ 


36*    Gor.  1.     If  (7  denote  the  given  chord,  c  the  chord  of 
half  the  arc,  the  equation  becomes 


=  y  2  i?2  _  2  i?  Jr^ 


4 


=  V2  R^  —Rsj4.  R^ 


G' 


37.    Gor.  2.     If  the  diameter  D,  that  is,  2  R,  is  unity,  the 
equation  in  (36)  becomes 

c  =  Vi  -  i  sjr^^G'^ 


BOOK  III.  59 

PROPOSITION   XIII.  • 

PROBLEM. 

38t    To  find  the  arithmetical  value  of  the  constant  n. 

From  (30)  C  =  7rD;  ifZ)  =  l,  this  equation  becomes  C  =  tt. 
If  then  we  can  find  the  circumference  of  a  circle  whose  diame- 
ter is  unity,  we  shall  have  the  value  of  tt. 

If  the  diameter  is  unity,  radius  is  one  half,  and  the  side  of  a 
regular  hexagon  inscribed  in  the  circle  is  one  half  (33),  and  the 
perimeter  of  the  hexagon  is  6  X  2  ^^^  ^• 

As  the  diameter  is  unity,  and  the  side  of  the  inscribed  hexa- 
gon one  half,  we  can  find  the  side  of  the  regular  inscribed  dodec- 
agon from  the  equation  in  (37) : 


=  V/.5  —  .433 

=  ^."067  =  .2588+ 
The     perimeter    of    the    inscribed     dodecagon     is    therefore 
12  X  .2588+ =3.105+. 

By  using  the  side  of  the  dodecagon  =  .2588+,  as  C,  or 
.067  =  C%  from  the  same  equation  we  can  find  the  side  of  a 
regular  inscribed  polygon  of  twenty-four  sides : 


-.483 
=  V^.0T7  =  .  13038 
The  perimeter  of  the  inscribed  polygon  of  twenty-four  sides  is 
therefore  24  X  .13038  =  3.129. 

By  continuing  this  process  we  approximate  to  the  circumfer- 
ence, that  is,  to  the  value  of  tt. 


60  PLANE  GEOMETRY. 

39.  Scholium.  By  other  more  expeditious  methods  the  value 
of  TT  has  been  found  accurately  to  two  hundred  and  fifty  places 
of  decimals.  For  practical  purposes  it  is  sufi&ciently  accurate 
to  call  7r  =  3.14159. 


PRACTICAL   QUESTIONS. 

1.  What  is  the  circumference  of  a  circle  whose  radius  is  10  feet  ?  '^« 

2.  What  is  the  diameter  of  a  circle  whose  circumference  is  57  rods  ? 

3.  What  is  the  area  of  a  circle  whose  radius  is  40  feet  ? 

4.  What  is  the  area  of  a  circle  whose  circumference  is  18  inches  ? 

5.  What  is  the  circumference  of  a  circle  whose  area  is  116  square  feet  ? 

6.  The  radii  of  two  concentric  circles  are  40  and  54  feet ;  what  is  the 
area  of  the  space  bounded  by  their  circumferences  ? 

7.  A  has  a  circular  lot  of  land  whose  diameter  is  95  rods,  and  B  a  simi- 
lar lot  whose  area  is  750  square  rods  ;  compare  these  lots. 

8.  What  is  the  difference  between  the  perimeters  of  two  lots  of  land  each 
containing  an  acre,  if  one  is  a  square  and  the  other  a  circle  ? 

9.  What  is  the  area  of  a  square  inscribed  in  a  circle  whose  area  is  a 
square  metre  ? 

10.  What  is  the  area  of  a  regular  hexagon  inscribed  in  a  circle  whose 
area  is  567  square  feet. 

11.  If  a  rope  an  inch  in  diameter  will  support  1,000  ^inds,  what  mjpist 
be  the  diameter  of  a  rope  of  like  material  to  support  4,  (M^  pounds  ?   .    j*' 

12.  If  a  pipe  an  inch  in  diameter  will  fill  a  cistern  in  25  minutes,  how 
long  will  it  take  a  pipe  5  inches  in  diameter  ? 

13.  If  a  pipe  an  inch  in  diameter  will  empty  a  cistern  in  an  hour,  how 
long  will  it  take  this  pipe  to  empty  the  cistern  if  there  is  another  pipe  one 
third  of  an  inch  in  diameter  through  which  the  fluid  runs  in  ? 

Ans.   67^  minutes. 

14.  If  a  pipe  3  inches  in  diameter  will  empty  a  cistern  in  3  hours,  how 
long  will  it  take  the  pipe  to  empty  the  cistern  if  there  are  3  other  pipes 
each  an  inch  in  diameter  through  which  the  fluid  runs  in. 

Ans.   4^  hours. 


BOOK  m. 


61 


EXERCISES. 

The  following  Theorejd!s,  depending  for  their  demon§tpatton  upon 
those  already  demonstrated,  ace  introd«ced  as  exercises  for  the  pupil. 
In  some  o^them  reiferenc.es"are  made  to  tlie  propositions  upon  which 
the  demonstraS^srfTJepends.  They  are  not  connected  with  the  prop- 
ositiojaeritt  tlje 'following  books,  and  can»i^e  omitted  if  thonirht  best. 

.  -  "'   ^  «•-<■* 

circumference.  . ,  .,,.J 


XiPr^EVsery. diameter  bisects  the  c^e  a^  the  circi 
41.     '     III  liikliilimM 1111^  llflllllir (     if 


two  p^n^.     (4.)     (I.  51.) 

42.  The  diametei^'lir^reektei^jJl^Ii,  anx.^fe 

43.  In  the  same  or  equal  circles,  when 
the  sum  of  the  arcs  is  less  than  a  circumfer- 
ence, the  greater  arc  is  subtended  by  the 
greater  chord;  and,  conversely,  the  greater   At 
chord  is  subtended  by  the  greater  arc. 

Draw  ^  a     (21.)     (1.47.) 
What  is  the  case  when  the  sum  of  the  arcs 
is  greater  than  a  circumference  ? 

44.  In  the  same  circle  equal  chords  are  equally  distant  from  the 
centre ;  and  of  two  unequal  chords  the  greater  is  nearer  the  centre. 

45.  The  shortest  and  the  longest  line  that  can  be  drawn  from  any 
point  to  a  given  circumference  lies  on  the  line  that  passes  from  the 
point  to  the  centre  of  the  circle. 

46.  Two  parallels  cutting  the  circumference  of  a  circle  intercept 
equal  arcs. 

47.  A  straight  line  perpendicular  to  a 
diameter  at  its  extremity  is  a  tangent  to  the 
circumference. 

Draw  GB.    (1.51.)  ^! 

48.  The  lines  joining  the  extremities  of 
two  diameters  are  parallel. 

49.  If  the  extremities  of  two  chords  are  joined,  the  vertical,  or 
opposite,  triangles  thus  formed  are  similar. 


62 


PLANE  GEOMETRY. 


50«  If  twQ  circumferences  cut  each  other,  the  chord  which  joins 
their  points  of  intersection  is  bisected  at  right  angles  by  the  Hne  join- 
ing their  centres.     (17.) 

51*  If  two  circumferences  touch  each  other,  their  centres  and 
point  of  contact  are  in  the  same  straight  hne,  perpendicular  to  the 
tangent  at  the  point  of  contact.     (47.) 

52»  The  distance  between  the  centres  of  two  circles  whose  cir- 
cumferences cut  one  another,  is  less  than  the  sum,  but  greater  than 
the  difference,  of  their  radii. 

53»  Every  angle  inscribed  in  a  segment  greater  than  a  semicircle 
is  acute ;  and  every  angle  inscribed  in  a  segment  less  than  a  semicir- 
cle is  obtuse.     (21.) 

54«    The  angle  made  by  a  tangent  and  a 
chord  is  measured  by  half  the  included  arc. 
Draw  the  diameter  A  B.     (47.)     (21.) 

55 1  The  angle  formed  by  two  chords  cut- 
ting each  other  within  the  circle  is  measured 
by  half  the  sum  of  the  arcs  intercepted  by  its 
sides  and  by  the  sides  of  its  vertical  angle. 

Join  B  C  (in  lower  figure).     (21.) 

56.  By  moving  the  point  of  intersection 
of  the  two  chords,  show  that  (14)  and  (21) 
can  be  deduced  from  (55). 

57 •  The  segments  of  two  chords  inter- 
secting within  a  circle  are  reciprocally  pro- 
portional ;  that  is,AE:  BE  =  ED:  EG. 

Join  AD,BG.     (21.)    (II.  20  ) 

58.  The  opposite  angles  of  a  quadrilateral  inscribed  in  a  circle  are 
supplementary.     (21.) 

59.  A  quadrilateral  whose  opposite  angles  are  supplementary,  and 
no  other,  can  be  mscribed  in  a  circle. 

60.  Lines  through  the  point  of  contact  »f  two  circumferences  that 
are  tangent  to  each  other  are  cut  proportionally  by  these  circumfer- 
ences.    (22.)     (II.  20.) 

61 .  The  area  of  a  sector  is  equal  to  half  the  product  of  its  arc  by 
the  radius  of  the  circle.     (31.) 


BOOK  III. 


63 


62*    Show  how  to  find  the  area  of  a  segment  of  a  circle. 

63*    The  area  of  a  circumscribed  polygon  is  equal  to  half  the  pro- 
duct of  its  perimeter  by  the  radius  of  the  circle. 

64*  A  tangent  is  a  mean  proportional 
between  a  secant  drawn  from  the  same 
point  and  the  part  of  the  secant  without 
the  circle. 

Join  A D,  DC.     (54 ;  21.)     (II.  57.) 

65*  The  angle  formed  by  two  secants, 
two  tangents,  or  a  secant  and  a  tangent 
cutting  each  other  without  the  circle,  is 
measured  by  half  the  difference  of  the  in- 
tercepted arcs. 

Join  CF.     (I.  39.)     (21.) 

66i  By  moving  the  point  of  intersec- 
tion, show  that  (21)  can  be  deduced  from 
(65).  Show  also  that  (46)  can  be  deduced 
from  (65). 

67»  Two  secants  drawn  from  the  same 
point  are  to  each  other  inversely  as  the 
parts  of  the  secants  without  the  circle. 

Join  GF,  D  G.     (21.)     (II.  57.) 

68.    Two  tangents  drawn  to  a  circumference  from  the  same  point 
are  equal. 

Join  B  E.     Figure  in  (66.)    (54.) 

69«  A  perpendicular  from  a  circumference 
to  the  diameter  is  a  mean  proportional  be- 
tween the  segments  of  the  diameter. 

Join  AB,BG.     (23.)     (II.  26.) 

70.    If  from  one  end  of  a  chord  a  diame- 
ter is  drawn,  and  from  the  other  end  a  per- 
pendicular to  this  diameter,  the  chord  is  a  mean  proportional  be- 
tween the  diameter  and  the  adjacent  segment  of  the  diameter. 

Join  A  B    (23.)   (11.25.) 

Tl»    The  sum  of  the  opposite  sides  of  a  circumscribed  quadrilat- 
eral is  equal  to  the  sum  of  the  other  two  sides.     (68.) 


BOOK  IV. 


GEOMETRY   OF  SPACE. 

PLANES  AND   THEIR  ANGLES. 

DEFINITIONS. 

1 ,  A  straight  line  is  perpendicular  to  a  plane  when  it  is  per- 
pendicular to  every  straight  line  of  the  plane  which  it  meets. 

Conversely,  the  plane,  in  this  case,  is  perpendicular  to  the 
line. 

The /oa^  of  the  perpendicular  is  the  point  in  which  it  meets 
the  plane. 

2.  A  line  and  a  plane  are  parallel  when  they  cannot  meet 
though  produced  indefinitely. 

3«  Two  planes  are  parallel  when  they  cannot  meet  though 
produced  indefinitely. 

THEOREM  I. 

4(    A  plane  is  determined, 

1  st.    By  a  straight  line  and  a  point  without  that  lin^  ; 
2d.    By  three  points  not  in  the  same  straight  line  ; 
3d.    By  two  intersecting  straight  lines. 

1st.  Let  the  plane  MN,  pass- 
ing through  the  line  A  B,  turn 
upon  this  line  as  an  axis  until  it 
contains  the  point  C ;  the  posi- 
tion of  the  plane  is  evidently  de- 
termined; for  if  it  is  turned  in 
either  direction  it  will  no  longer  contain  the  point  (7. 


BOOK  IV. 


65 


2d.  If  three  points,  A,  B,  C,  not  in  the  same  straight  line 
are  given,  any  two  of  them,  as  A  and  B,  may  be  joined  by  a 
straight  line ;  then  this  is  the  same  as  the  1st  case. 

3d.  If  two  intersecting  lines  A  B,  AC  are  given,  any  point, 
C,  out  of  the  line  A  B  can  be  taken  in  the  line  A  C ;  then  the 
plane  passing  through  the  line  A  B  and  the  point  C  contains 
the  two  lines  A  B  and  A  C,  and  is  determined  by  them. 

5t  Corollary.  The  intersection  of  two  planes  is  a  straight 
line  ;  for  the  intersection  cannot  contain  three  points  not  in  the 
same  straight  line,  since  only  one  plane  can  contain  three  such 
points. 

THEOREM  li. 


6*  Ohlique  lines  from  a  point  to  a  plane  equally  distant  from 
the  perpendicular  are  equal ;  and  of  two  ohlique  lines  unequally 
distant  from  the  perpendicular^  the  more  remote  is  the  greater. 

Let  AC,  A  D  be  oblique  lines 
drawn  to  the  plane  if  iV  at  equal 
distances  from  the  perpendicular 
AB: 

1st.  A  C=  AB;  {or  the  trian- 
gles ^^(7,  ABB  are  equal  (I.  40). 

2d.  Let  ^  i^  be  more  remote. 
From  BF  cut  off  BE  =  BB 
and  draw  A  E ;  then  A  F  "^  AE 
(L  51);  and  AE  =  AB  =  AC;  therefore  AF>  ABor  AC. 

7.  Cor.  1.  Conversely,  equal  oblique  lines  from  a  point  to  a 
plane  are  equally  distant,  from  the  perpendicular;  therefore 
they  meet  the  plane  in  the  circumference  of  a  circle  whose  cen- 
tre is  the  foot  of  the  perpendicular.  Of  two  unequal  lines  the 
greater  is  more  remote  from  the  perpendicular. 

8.  Cor.  2.  The  perpendicular  is  the  shortest  distance  from 
a  point  to  a  plane. 


d 

¥ 

\ 

/'     1 

-• 

'•       /  B 

\\e\^ 

/  / 

\ 

A/ 

^ 

N 


66  GEOMETKY   OF   SPACE. 

THEOREM  III. 

9.  The  intersections  of  two  parallel  planes  vrith  a  third  plane 
are  parallel. 

Let  A  B  and  C  D  he  the  intersec-  B 

tions  of  the  plane  A  D  with  the  par-  \        /\  \ 

allel  planes  M  N  and  P  Q;  then  A  B  \    /     \  \ 

and  CD  are  parallel.  \  \ 

For  the  lines  A  B  and  G  D  cannot  \  \^ 

meet    though    produced    indefinitely,  \  \         /  \ 

since  the  planes  M  N  and  P  Q  in  which  \ \/       \ 

they  are  cannot  meet ;  and  they  are  in  C  ^ 

the  same  plane  A  D ;  therefore  they  are  parallel. 

10.  Corollary.  Parallels  intercepted  between  parallel  planes 
are  equal.  For  the  opposite  sides  of  the  quadrilateral  A  D  be- 
ing parallel,  the  figure  is  a  parallelogram ;  therefore  AC  =.B  D. 

THEOREM  IV. 

lit  If  two  angles  not  in  the  same  plane  have  their  sides  paral- 
lel and  similarly  situated,  the  angles  are  equal  and  their  planes 
parallel. 


I 


Let  ABC  and  DE F  he  two  angles 

A/" 
in  the  planes  M  iV  and  P  Q,  having 

their  sides  A  B,  B  C  respectively  paral- 
lel to  JDUf  E  F,  and  similarly  situated ; 
then 

1st.  Since  BA  has  the  same  direc-     p 

tion   as  E  D,   and   B  C  the   same   as        \  ^ 

E  F,  the  difference  of  direction  of  ^  ^  \  D^  ^^ 

and  B  C  must  be  the  same  as  the  differ- 
ence of  direction  of  E  D  and  E  F;  that  is,  angle  B  =  angle  E. 

2d.  The  planes  of  these  angles  are  parallel.  For,  since  two 
intersecting  lines  determine  a  plane  (4),  the  plane  of  the  lines 
A  B  and  B  C  must  be  parallel  to  the  plane  of  the  lines  D  E  and 
E  F,  Sis  AB  and  B  C  are  respectively  parallel  to  D  E  and  E  F. 


BOOK  IV. 


67 


THEOREM  V. 

12.    //  two  straight  lines  are  ciU  hy  parallel  planes,  they  are 
divided  proportionally. 


Vc^\ 

\  , 

V 

_  Vjr\ 

M 

i-4 

N 


Let  A  B  and  C  D  he  cut  by  the  parallel   M 
planes  M  N,  PQ,  and  R  S,  in  the  points 
A,  E,  B,  and  C,  F,  D-,  then 

AE:EB  =  GF:FD  ^ 

For,  drawing  A  D  meeting  the  plane  P  Q 
in  6r,  the  plane  of  the  lines  A  B  and  AD    ^ 
cuts  the  parallel  planes  PQ  and  ES  in 
E  G  and  B  D  ;  therefore  E  G  and  B  D  are 
parallel  (9),  and  we  have  (XL  16) 

AE  :EB=:AG  :  GD 
The  plane  of  the  lines  A  D  and  G D  cuts  the  parallel  planes 
MN  and  PQ'mAG  and  GF;  therefore  AG'm  parallel  to  GF', 
and  we  have 

AG  :GD=zGF  :FD 
Hence  we  have  (Pn.  11) 

AE  :EB=:GF  '.FD 


bo  GEOMETRY   OF   SPACE. 

EXERCISES. 

The  following  Theorems,  depending  for  their  demonstration  upon 
those  already  demonstrated,  are  introduced  as  exercises  for  the  pupil. 
In  some  of  them  references  are  made  to  the  propositions  upon  which 
the  demonstration  depends.  They  are  not  connected  with  the  prop- 
ositions in  the  following  books,  and  can  be  omitted  if  thought  best. 

13*  An  infinite  number  of  planes  can  pass  through  a  given 
line.     (4.) 

14«    There  can  be  but  one  perpendicular  from  a  point  to  a  plane. 

15«  A  line  perpendicular  to  each  of  two  lines  at  their  point  of 
intersection  is  perpendicular  to  the  plane  of  these  lines.    (4.)    (I.  76.) 

16(    Parallel  lines  are  equally  inclined  to  the  same  plane. 

17«    State  the  converse  of  (16).     Is  it  true? 

18*  Lines  parallel  to  a  line  in  a  given  plane  are  parallel  to  the 
plane. 

19.    State  the  converse  of  (18).     Is  it  true  ? 

20*    Parallel  planes  are  equally  inclined  to  the  same  straight  line. 

21 1    State  the  converse  of  (20).     Is  it  true? 


BOOK    V. 


POLYEDKONS. 

DEFINITIONS. 

1.   A  Polyedron  is  a  solid  bounded  by  planes. 
The  bounding  planes  are  called  faces ;   their  intersections, 
edges  ;  the  intersections  of  the  edges,  vertices. 

2t  The  Volume  of  a  solid  is  the  measure  of  its  magnitude. 
It  is  expressed  in  units  which  represent  the  number  of  times  it 
contains  the  cubical  unit  taken  as  a  standard. 

3*    Equivalent  Solids  are  those  which  are  equal  in  volume. 

4*  Similar  Solids  are  those  whose  homologous  lines  have  a 
constant  ratio.  {Corollary.)  It  follows  that  similar  solids  are 
bounded  by  the  same  number  of  similar  polygons  similarly 
situated. 


PRISMS  AND   CYLINDERS. 

5*  A  Prism  is  a  polyedron  two  of  whose  faces 
are  equal  polygons  having  their  homologous  sides 
parallel,  and  the  other  faces  parallelograms. 
(Corollary.)     The  lateral  edges  are  equal. 

The  equal  parallel  polygons  are  called  bases; 
as  ^^  and  CD. 

6.  The  Altitude  of  a  prism  is  the  perpendic- 
ular distance  between  its  bases ;  SiS  B  F. 


70 


SOLID   GEOMETRY. 


7.    A  Right  Prism  is  one  whose  other  faces  are 


perpendicular  to  its  bases, 
faces  are  rectangles. 


{Corollary  )   Its  lateral 


8.  A  prism  is  called  triangular,  quadrangular,  or 
pentagonal,  according  as  its  base  is  a  triangle,  a 
quadrangle,  or  a  pentagon;   and  so  on. 

9.  A  Parallelopiped  is  a  prism  whose  bases  are 
parallelograms.  {Corollary.)  It  follows  that  all  its 
faces  are  parallelograms. 

A  Right  Parallelopiped  is  a  right  prism  whose 
bases  are  parallelograms. 

10.  A  Rectangular  Parallelopiped  is  a  right 
parallelepiped  whose  bases  are  rectangles. 

{Corollary.)   All  its  faces  are  rectangles. 

11.  A  Cube  is  a  parallelopiped  whose  faces  are 
all  squares.  {Corollary.)  It  follows  that  its  faces 
are  all  equal,  and  the  parallelopiped  rectangular. 

.  12.  A  Cylinder  is  a  right  prism  whose  bases  are 
regular  polygons  of  an  infinite  number  of  sides, 
that  is,  whose  bases  are  circles.  A  cylinder  can 
be  described  by  the  revolution  of  a  rectangle  about 
one  of  its  sides  which  remains  fixed.  The  side  oppo- 
site the  fixed  side  describes  the  convex  surface,  and 
the  other  two  sides  the  two  circular  bases.  Thus 
the  rectangle  AB  C D  revolving  about  B  C  would 
describe  the  cylinder,  the  side  A  D  the  convex  sur- 
face, and  AB,  D  C the  circular  bases. 


13.  The  Axis  of  a  cylinder  is  the  straight  line  joining  the 
centres  of  the  two  bases ;  or  it  is  the  fixed  side  of  the  rectangle 
whose  revolution  describes  the  cylinder ;  as  5  (7. 


BOOK  V. 


71 


THEOREM  I. 

14,    The  convex  surface  of  a  right  prism  is  equal  to  the  perime- 
ter of  its  6ase  multiplied  hy  its  altitude. 

Let  ^  ^  be  a  right  prism ;  its  convex  surface 
is  equal  to  FG-\-GH-\-HI^IK-\-KF 
multiplied  by  its  altitude  A  F. 

For  the  convex  surface  is  equal  to  the  sum 
of  the  rectangles  AG,  BH,  CI,  &c.  The  area 
of  the  rectangle  AG  =  FG  X  ^F;  the  area 
of  BH=zGHXBG]  of  CI=HIx  C H -, 
and  so  on.  But  the  edges  AF,  B  G,  C  H,  &c. 
are  equal  to  each  other  and  to  the  altitude  of  the  prism ;  and 
the  bases  of  these  rectangles  together  form  the  perimeter  of  the 
prism.  Therefore  the  sum  of  these  rectangles,  that  is,  the  con- 
vex surface  of  the  right  prism,  is  equal  to  the  perimeter  of  its 
base  multiplied  by  its  altitude. 

15*  Corollary.  As  a  cylinder  is  a  right  prism 
(12),  this  demonstration  includes  the  cylinder.  If, 
then,  R  =  the  radius  of  the  base,  and  A  =  the  alti- 
tude of  a  cylinder,  the  convex  surface  =  2  tt  BA. 


THEOREM  II. 

16t    The  sections  of  a  prism  made  hy  parallel  planes  are  equal 
polygons. 

Let  the  prism  ^  JjTbe  intersected  by  the  par- 
allel planes  LN  and  Q  S ;  then  LN  and  ^*S' 
are  equal  polygons.  For  LM,  M N,  NO,  &c. 
are  respectively  parallel  to  Q R,  R S,  ST,  &c. 
(IV.  9),  and  similarly  situated ;  therefore  the 
angles  L,  M,  N,  0,  P  are  respectively  equal  to 
the  angles  Q,  R,  S,  T,  U  (IV.  11);  and  the 
polygons  LN  and  QS  are  mutually  equiangu- 
lar.    Also  the  sides  LM,  M N^  NO,  &c.  are 


72 


SOLID  GEOMETRY. 


respectively  equal  to  Q R,  RS,  ST,  &c.  (I.  62).  Therefore 
the  polygons,  being  mutually  equiangular  and  equilateral,  are 
equal  (11.  6). 

17t    Cor.  1.     A  section  made  by  a  plane  parallel  to  the  base 
is  equal  to  the  base. 

18.    Cor.  2.     A  section  of  a  cylinder  made  by  a  plane  paral- 
lel to  the  base  is  a  circle  equal  to  the  base. 


THEOREM  III. 


19.    Prisms 
equivalent. 


having  equivalent   bases  and  equal  altitudes  are 


Let  A  C  and  FH  be  two  prisms 
having  equal  altitudes  and  their 
bases  B  O,  G  H  equivalent ;  the 
prisms  are  equivalent. 

Let  DU  and  IK  he  sections 
made  by  planes  respectively  par- 
allel to  the  bases  BG  and  GH; 
these  sections  are  respectively 
equal  to  the  bases  (17);  there- 
fore the  section  DE  m  equivalent 
to  IK,  at  whatever  distance  from 

the  base  either  may  be.  If,  therefore,  the  planes  of  these  sec- 
tions move,  remaining  always  parallel  to  the  bases,  as  the  sec- 
tions will  always  be  equivalent,  it  is  evident  that  in  moving 
over  an  equal  length  of  altitude  the  sections  will  move  over 
equal  volumes ;  therefore,  as  the  altitudes  are  equal,  the  prisms 
are  equivalent. 

20.    Corollary.    Any  prism  is  therefore  equivalent  to  a  rec- 
tangular prism  having  an  equivalent  base  and  an  equal  altitude. 


BOOK  V.  73 


THEOREM  IV.  ^ 

21 .  The  volume  of  a  rectangular  parallelopiped  is  equal  to  the 
product  of  its  three  dimensions. 

Let  -4  2>  be  a  rectangular  parallelopiped ;  then  its  volume  is 
equal  toBCxBExBA.  Suppose 
B  Fj  the  linear  unit,  is  contained  in  B  C 
four  times,  in  B  E  five  times,  and  in  B  A 
seven  times ;  then  dividing  B  C,  B  E,  B  A 
respectively  into  four,  five,  and  seven 
equal  parts,  and  passing  planes  through 
the  several  points  of  division  parallel 
to  the  sides  of  the  parallelopiped,  there 
will  be  formed  a  number  of  cubes  equal 

to  each  other  (19),  and  each  equal  to  the  cube  whose  edge 
is  the  linear  unit.  It  is  evident  also  that  the  whole  number 
of  cubes  is  equal  to  the  product  of  the  three  dimensions,  or 
4  X  S  X  7  =  140.  This  demonstration  is  applicable,  what- 
ever the  number  of  units  in  the  linear  dimensions  may  be. 
Therefore  the  volume  of  a  rectangular  parallelopiped  is  equal 
to  the  product  of  its  three  dimensions. 

22t  Scholium.  If  the  three  dimensions  are  incommensur- 
able, the  linear  unit  can  be  taken  infinitely  small,  that  is,  so 
small  that  the  remainder  will  be  infinitesimal  and  can  be  neg- 
lected. 

23*  Cor.  1.  As  the  base  is  equal  to  B  C  X  B  E,  the  volume 
of  a  rectangular  parallelopiped  is  equal  to  the  product  of  its 
base  by  its  altitude. 

24*  Cor.  2.  The  volume  of  a  cube  is  equal  to  the  cube  of 
its  edge. 


SOLID  GEOMETRY. 


THEOREM  V. 


25*  The  volume  of  any  prism  is  equal  to  the  product  of  its  base 
by  its  altitude. 

For  any  prism  is  equivalent  to  a  rectangular  parallelopiped, 
having  an  equivalent  base  and  the  same  altitude  (20) ;  and  the 
volume  of  the  equivalent  rectangular  parallelopiped  is  equal  to 
the  product  of  its  base  by  its  altitude ;  therefore  the  volume  of 
any  prism  is  equal  to  the  product  of  its  base  by  its  altitude. 

26.  Corollary.  As  a  cylinder  is  a  right  prism,  this  demon- 
stration includes  the  cylinder.  If,  therefore,  R  =  the  radius  of 
base,  A  =  the  altitude,  and  V  ==.  the  volume  of  a  cylinder, 


THEOREM  VI. 

27.    Similar  prisms  are  as  the  cubes  of  their  homologous  lines. 

Let  A  D  and  B  R  he  similar 
prisms  whose  altitudes  are  IK  and 
MN.  Let  V  represent  the  vol- 
ume of  A  JD,  and  v  the  volume  of 
Uff;  then 

V:v  =  IK^  :  MN^  =  AC^:  UG^ 
=  CO''  :  GF'' 

For  (25)   V=CD  X  IK  and 
v=GII  X  MN,  therefore 
V'.v=CDXlK'.  GHXMN 

But(n.  31)  CD  :GHz=CO'' 

and  (4)  IK:MN=CO\ 

Multiplying  the  last  two  proportions  together  we  have 

CDXIK.GHX  MN=CO^:GI^ 
therefore  (Pn.  11)  V -.  v  =  C  O"" -.  G  P"" 

But  in  similar  solids  homologous  lines  have  a  constant  ratio 
(4) ;  therefore  F  :  v  as  the  cubes  of  any  homologous  lines. 


BOOK  V. 


75 


PYRAMIDS   AND   CONES. 


DEFINITIONS. 


28.  A  Pyramid  is  a  polyedron  bounded  by  a  polygon  called 
the  base,  and  by  triangular  planes  meeting  at  a  common  point 
called  the  vertex. 

29.  A  pyramid  is  called  triangular,  quad-  ^ 
rangular,  'pentagonal^  according  as  its  base 
is  a  triangle,  a  quadrangle,  or  a  pentagon  ; 
and  so  on. 

30.  The  Altitude  of  a  pyramid  is  the 
perpendicular  distance  from  its  vertex  to 
its  base ;  as  ^  ^. 

31 .  A  Bight  Pyramid  is  one  whose  base 
is  a  regular  polygon  and  in  which  the  per- 
pendicular from  the  vertex  passes  through  the  centre  of  the  base. 

32.  The  Slant  Height  of  a  right  pyramid  is  the  perpendicu- 
lar distance  from  the  vertex  to  the  base  of  any  one  of  its  lateral 
faces  ;  as  ^  C. 

33.  A  Cone  is  a  right  pyramid  whose 
base  is  a  regular  polygon  of  an-  infinite 
number  of  sides,  that  is,  whose  base  is  a 
circle.  A  cone  can  be  described  by  the  rev- 
olution of  a  right  triangle  about  one  of  its 
sides  which  remains  fixed.  The  other  side 
describes  the  circular  base,  and  the  hypoth- 
enuse  the  convex  surface.  Thus  the  right 
triangle  ABC  revolving  about  A  B  would 
describe. the  cone,  BC  the  base,, and  the  h^^pothenuse  A  C  the 
convex  surface. 

34.  The  Axis  of  a  cone  is  the  line  from  the  vertex  to  the 
centre  of  the  base  ;  or  it  is  the  fixed  side  of  the  right  triangle 
whose  revolution  describes  the  cone  ;  as  A  B,  » 


76 


SOLID    GEOMETRY. 


3.5*    Corollary.     The  axis  of  a  cone  is  perpendicular  to  the 
base,  and  is  therefore  the  altitude  of  the  cone. 


36*  A  Frustum  of  a  pyramid  is  a  part  of 
the  pyramid  included  between  the  base  and 
a  plane  cutting  the  pyramid  parallel  to  the 
base ;  aa  D  £!. 

37 •  The  Altitude  of  a  frustum  is  the  per- 
pendicular distance  between  the  two  parallel  planes  or  bases  ; 

38t    The  Slant  Height  of  a  frustum  of  a  right  pyramid  is 
the  perpendicular  distance  between  the  parallel  edges  of  the 
as  6^  a 


f 


THEOREM  YII. 


39*  If  «  pyramid  is  cut  by  a  plane  parallel  to  its  hose, 
1st.  The  edges  and  altitude  are  divided  proportionally  ; 
2d.    The  section  is  a  polygon  similar  to  the  base. 

Let  A-BCDEF  be  a  pyramid  whose  al-  ^ 

titude  is  A  N,  cut  by  a  plane  G I  parallel 
to  the  base  ;  then 

1  St.  The  edges  and  the  altitude  are  di- 
vided proportionally. 

For  suppose  a  plane  passed  through  the 
vertex  A  parallel  to  the  base;  then  the 
edges  and  altitude,  being  cut  by  three 
parallel  planes,  are  divided  proportion- 
ally (IV.  12),  and  we  have 

AB'.AG  =  A  C  -.AH^AD  '.AI=AN'.AM 

2d.    The  section  GI  '\%  similar  to  the  base  BD. 

For  the  sides  of  G I  are  respectively  parallel  to  the  sides  of 
B  D  (IV.  9),  and  similarly  situated ;  therefore  the  polygons  GI, 
BD  are  mutually  equiangular.     Also,  as  (rX  is  parallel  to  BF, 


BOOK  V.  77 

and  LK  to  F  E,  the  triangles  ^  ^i^and  AG  L  qxq  similar,  and 
the  triangles  A  F E  and  ALK )  therefore 

GL'.BF=AL'.AF,2.ndiLK:FE=AL'.AF 

Therefore  GL  :  BF=zLK:  FE 

In  the  same  manner  we  should  find 

LK:FE=zKI'.ED  =  IH'.DC,kG, 

Therefore  the  polygons  G I  and  B  D  are  similar  (II.  19). 

40t  Corollary.  A  section  of  a  cone  made  by  a  plane  parallel 
to  the  base  is  a  circle. 

y^  THEOREM  VIII. 

41*  The  convex  surface  of  a  right  pyramid  is  equal  to  the 
perimeter  of  its  base  multiplied  by  half  its  slant  height. 

A 

Let  ^-5  GB  EF  be  a  right  pyramid  whose 
slant  height  \^  AH\  its  convex  surface  is 
equal  to  B  G  ■\- G  B -\- B  E -\- E  F -^^  F  B 
multiplied  by  half  of  A  H. 

The  edges  AB,  AG,  AD,  AE,  A  F,  be- 
ing equally  distant  from  the  perpendicular 
A  J^  (11.  34),  are  equal  (IV.  6) ;  and  the 
bases  BG,  G D,  BE,  &c.  are  equal;  there- 
fore the  isosceles  triangles  ABG,  AGB, 
ABE,  (fee.  are  all  equal  (I.  48) ;   and  their 
altitudes  are  equal.     The  area  oi  ABG  \^  BG  X  J  ^ //  (II.  11); 
oi  AG  B'mG  By,  J  ^  -^ ;  and  so  on.     Therefore  the  sum  of  the 
areas  of  these  triangles,  that  is,  the  convex  surface  of  the  right 
pyramid,  m{BG-\-GD-{-DE-^EF+FB)\AH. 

42*  Gorollary.  As  a  cone  is  a  right  pyramid  (33),  this  dem- 
onstration includes  the  cone.  If,  therefore,  R  =  the  radius  of 
the  base,  and  aS'  =  the  slant  height  of  a  cone, 

its  convex  surface  =  2  n  R\  S  =:.  n  R  S 

If  a  plane  parallel  to  the  base  and  bisecting  the  altitude  be 


78  SOLID   GEOMETRY. 

drawn,  as  the  section  will  be  a  circle  (40)  with  a  radius  and  cir- 
cumference one  half  the  radius  and  circumference  of  the  base, 
therefore,  if  r'  =  the  radius  of  this  section, 

the  convex  surface  =  2irr'  S 


,    '  THEOREM  IX. 

43t  The  convex  surface  of  a  frustum  of  a  right  pyramid  is 
equal  to  the  sum  of  the  perimeter  of  its  two  bases  multiplied  hy 
half  its  slant  height. 

Let  G  D  he  the  frustum  of  a  right  pyra-  B 

mid ;  its  convex  surface  is  equal  to  G  H  -\-  /^   i    "--  / 

HI-\-IK+KL  -\-LG+  BC+CD  fYpf\ 

+  Di:-\-BF+  FB  multiplied  hy  half  bU'T'^A^A 

MN.  \             V'^D 

N\i  \  X 

The  lateral  faces  of  a  frustum  of  a  right     ^       ^ Y 

pyramid  are  equal  trapezoids  (39  ;  11.  6) ; 
and  their  altitudes  are  all  equal.  The  area  oi  G  G  (II.  14)  is 
(GH+  BG)  X  IMN;  of  HD  is  {HI  +  G D)  X  \MN', 
and  so  on.  Therefore  the  sum  of  the  areas  of  these  trapezoids, 
that  is,  the  convex  surface  of  the  frustum  of  the  right  pyra- 
mid, is  GH-\-HI^IK  -\-  KL^  LG  -\-BG  +  GD  + 
DE-^-  EF  -\-  FB  multiplied  by  half  M N. 

44*  Gor.  1.  If  the  frustum  is  cut  by  a  plane  parallel  to  its 
two  bases,  and  at  equal  distances  from  each  base,  this  plane 
will  bisect  the  edges  G B^  HG,  ID,  &c.  (39) ;  and  the  area  of 
each  trapezoid  is  equal  to  its  altitude  multiplied  by  the  line 
joining  the  middle  points  of  the  sides  which  are  not  parallel 
(II.  15).  Therefore  the  convex  surface  of  a  frustum  of  a  right 
pyramid  is  equal  to  the  perimeter  of  a  section  midway  between 
the  bases  multiplied  by  its  slant  height. 

45*  Cor.  2.  As  a  cone  is  a  right  pyramid  (33),  this  demon- 
stration includes  the  frustum  of  a  cone.     If,  therefore,  R  and 


BOOK   V. 


79 


r  =  the  radii  of  the  two  bases  of  the  frustum  of  a  cone,  and 

AS'=its  slant  height, 
its  convex  surface  =z(2irE-{-2irr)^S={irIi-\-irr)S 
If  r'  =  the  radius  of  a  section  midway  between  and  parallel 

to  the  bases, 

the  convex  surface  =  2irr'  S 


^ 


THEOREM  X. 


40*  If  two  pyramids  having  equal  altitudes  are  cut  by  planes 
parallel  to  their  bases  and  at  equal  distances  from  their  vertices^  the 
sections  are  to  each  other  as  their  bases. 


Let  A-BC DBF Sind 
G-UIK  be  two  pyra- 
mids of  equal  altitudes 
AT,  GW,  cut  by  the 
planes  LM N  0 P  and 
^i?AS' parallel  respectively 
to  the  bases  and  at  equal 
distances  from  the  vertices 
A  and  G,  then 


LMNOP:QRS=:BCDEF.HIK 
For  as  the  polygons  LMNOP  and  BCDEF  are  similar  (39) 
LMNOP  :  BCDEF  =  LP' :  BF^  =  aV  :  AB^=  AV^  :  If* 
In  like  manner 

QMS:  HIK=z  G'y''  :G~W^ 
But  SLS  AV  =  G  Y  ?ind  A  T  =  GW 

til  GrGiOTP 

LMNOP  \BCDEF=QRS',HIK 

or  (Pn.  16) 

LMNOP '.QRS=BCDEF '.HIK 

47i  Corollary.  If  two  pyramids  have  equal  altitudes  and 
equivalent  bases,  sections  made  by  planes  parallel  to  their  bases 
and  at  equal  distances  from  their  vertices  are  equivalent. 


/, 


80 


^ 


SOLID   GEOMETRY. 
THEOREM  XI. 


48«    Pyramids  having  equivalent  bases  and  the  same  altitude 
are  equivalent. 

Let  A-BC DBF  sind 
G-H I K  be  pyramids 
having  equivalent  bases 
and  equal  altitudes ;  then 
the  two  pyramids  are 
equivalent. 

For,  if  at  equal  dis- 
tances from  the  vertex 
sections    are   formed    by 

planes  parallel  respective-  K 

ly  to  their  bases,  these  sections  are  equivalent  (47).  If  now  the 
planes  forming  these  sections  be  supposed  to  move,  remaining 
always  parallel  to  the  bases,  and  each  keeping  the  same  distance 
from^  the  vertex  as  the  other,  these  sections,  always  being  equiv- 
alent to  each  other,  will  move  over  equal  volumes  ;  therefore,  as 
the  altitudes  are  equal,  the  pyramids  must  be  equivalent. 


THEOREM  Xn. 

49.    A  triangular  •pyramid  is  one  third  of  a  triangular  prism 
of  the  same  base  and  altitude. 

Let  C-D  E  Fhe  a  triangular  pyramid  and 
A  B  C-D  E  F  be  a  triangular  prism  on  the 
same  base  D  E  F -,  then  C-D  E  F  is  one 
ihirdio^  ABC-DEF, 

Taking  away  the  pyramid  C-D  E F  i\iQVQ 
remains  the  quadrangular  pyramid  whose  ver- 
tex is  C  and  base  the  parallelogram  ABED. 
Through  the  points  A,  C,  E  pass  a  plane  ;  it 
will  divide  the  pyramid  C-A  BED  into  two  triangular  pyra- 
mids, which  are  equivalent  to  each  other  (48),  since  their  bases 
are  halves  of  the  parallelogram  ABED,  and  they  have  the 


BOOK    V.         /  81 

same  altitude,  the-  perpendicular  from  their  vertex  C  to  the 
base  ABED.  But  the  pyramid  C-A  B  E,  that  is,  E-A  B  C, 
is  equivalent  to  the  pyramid  C-DEF,  as  they  have  equal 
bases  ABC  and  D  E  F,  and  the  same  altitude  (48).  Therefore 
the  three  pyramids  are  equivalent  and  the  given  pyramid  is  one 
third  of  the  prism. 

50.  Corollary.  The  volume  of  a  triangular  pyramid  is  equal 
to  one  third  the  product  of  its  base  by  its  altitude. 

THEOREM  XIII. 

51*  The  volume  of  any  pyramid  is  equal  to  one  third  of  the 
product  of  its  base  hy  its  altitude. 

Let  A-BC D E F  be  any  pyramid;  its 
volume  is  equal  to  one  third  the  product  of 
its  base  BCBEFhy  its  altitude  A  K 

Planes  passing  through  the  vertex  A  and 
the  diagonals  of  the  base  B £>,  BE,  will 
divide  the  pyramid  into  triangular  pyramids 
whose  bases  together  compose  the  base  of 
the  given  pyramid  and  which  have  as  their 
common  altitude  A  N,  the  altitude  of  the 
given  pyramid.  The  volume  of  the  given 
pyramid  is  equal  to  the  sum  of  the  volumes  of  the  several  tri- 
angular pyramids,  which  is  equal  to  one  third  of  the  sum  of 
their  bases  multiplied  by  their  common  altitude;  that  is,  is 
equal  to  one  third  of  the  product  of  the  base  BCD  FE  by  the 
altitude  A  N. 

52%  Cor.  1.  As  a  cone  is  a  right  pyramid  (33),  this  demon- 
stration includes  the  cone.  A  cone,  therefore,  is  one  third  of 
a  cylinder,  or  of  any  prism,  of  equivalent  base  and  the  same 
altitude.  If  7?  =  radius  of  the  base,  A  =  the  altitude,  and 
V  =  the  volume  of  a  cone,  V  =  ^n  B^A. 

53«  Cor.  2.  The  ratio  of  similar  pyramids  to  one  another  is 
the  same  as  that  of  similar  prisms ;  that  is,  as  the  cubes  of 
homologous  lines. 


82  SOLID   GEOMETRY. 

THE   SPHERE. 
DEFINITIONS. 

54 •  A  Sphere  is  a  solid  bounded  by  a  curved  surface,  of 
which  every  point  is  equally  distant  from  a  point  within  called 
the  centre,  A  sphere  can  be  described  by  the  revolution  of  a 
semicircle  about  its  diameter  which  remains  fixed. 

55i  The  Radius  of  a  sphere  is  the  straight  line  from  the  cen- 
tre to  any  point  of  the  surface. 

56t  The  Diameter  of  a  sphere  is  a  straight  line  passing 
through  the  centre  and  terminating  at  either  end  at  the  surface. 

57.  Corollary.  All  the  radii  of  a  sphere  are  equal ;  all  the 
diameters  are  equal,  and  each  is  double  the  radius. 


/ 


I  THEOREM  XIV. 

58.    Every  section  of  a  sphere  made  hy  a  plane  is  a  circle. 


Let  ABB  be  a  section  made  by  a 
plane  cutting  the  sphere  whose  centre  is 
G  \  then  is  J  ^  i)  a  circle. 

Draw  C  E  perpendicular  to  the  plane, 
and  to  the  points  A,  D,  F,  where  the 
plane  cuts  the  surface  of  the  sphere, 
draw  CA,  CD,  C F.  As  C A,  CD, 
C  F  are  radii  of  the  sphere  they  are 
equal,  and  are  therefore  equally  distant  from  the  foot  of  the 
perpendicular  C E  (IV.  7).  Therefore  E A,  ED,  EF  are 
equal,  and  the  section  A  B  D  is  a  circle  whose  centre  is  E. 

59.  Corollary.  If  the  section  passes  through  the  centre  of 
the  sphere,  its  radius  will  be  the  radius  of  the  sphere. 

60.  Definition.  A  section  made  by  a  plane  passing  through 
the  centre  of  a  sphere  is  called  a  great  circle.  A  section  made 
by  a  plane  not  passing  through  the  centre  is  called  a  small  circle. 


BOOK  V. 


83 


THEOREM  XV. 


61  •  The  surface  of  a  sphere  is  equal  to  the  product  of  its  diam- 
eter hy  the  circumference  of  a  great  circle. 

Let  ABCDEF  be  the  semicircle  by  whose 
revolution  about  the  diameter  A  F,  the  sphere 
may  be  described;  then  the  surface  of  the 
sphere  is  equal  to  the  diameter  A  F  multi- 
plied by  the  circumference  of  the  circle  whose 
radius  is  6^  ^4,  or  =  ^  ^  X  circ.  G  A. 

Let  A  B  C  D  E  F\)Q  a  regular  semi-decagon 
inscribed  in  the  semicircle.  Draw  G  0  per- 
pendicular to  one  of  its  sides,  as  B  C. 

Draw  B  K^  OP,  C  L,  D  M,  EN  perpendicular  to  the  diame- 
ter A  F,  and  B  H  perpendicular  to  G  L.  The  surface  described 
by  ^  C  is  the  convex  surface  of  the  frustum  of  a  cone,  and  is 
equal  to  BG  X  circ.  PO  (45).  But  the  triangles  ^(7 i7  and 
P  0  G  are  similar  (IL  21) ;  therefore 

BG  :BHoyKLz=GO  .  P  0 
or  (IIL  28)  BG  :  KL  =  circ.  G 0  :  circ.  P 0 

.'.  BGX  circ.  PG  —  KLX  circ.  G  G 
That  is,  the  surface  described  hy  B  G  is  equal  to  the  altitude 
KL  multiplied  by  circ.  G  0,  or  by  the  circumference  of  the  cir- 
cle inscribed  in  the  polygon.  In  like  manner  it  can  be  proved 
that  the  surfaces  described  by  ^  ^,  G  B,  D  E,  and  E  F  are 
respectively  equal  to  their  altitudes  A  K,  LM,  M  N,  and  N  F 
multiplied  by  circ.  G  0.  Therefore  the  entire  surface  described 
by  the  semi-polygon  will  be  equal  to 
{AK^KL  +  LM  +  MN-\-NF)c\rc.  GO  =  AFX  circ.  GO 

This  demonstration  is  true,  whatever  the  number  of  sides  of 
the  semi-polygon ;  it  is  true,  therefore,  if  the  number  of  sides 
is  infinite,  in  which  case  the  semi-polygon  would  coincide  with 
the  semicircle ;  and  the  surface  described  by  the  semi-polygon 
would  be  the  surface  of  the  sphere,  and  the  radius  of  the  in- 


84  SOLID   GEOMETRY. 

scribed  polygon  would  be  the  radius  of  the  sphere.     Therefore 
we  have  the  surface  of  the  sphere  equal  to 
^1  i^  X  circ.  G  A 

62*  Corollary.  Let  S  =  the  surface  of  the  sphere,  C  =  the 
circumference,  Ji  =  the  radius,  J)  =  the  diameter,  then  we 
have  (III.  30)  (7  =  2  tt  7?,  or  tt  /> 

Therefore  S=2'itR  X  2  i?  =  i'lrR^  or  tt  D^ 

That  is,  the  surface  of  a  sphere  is  equal  to  the  square  of  its  diame- 
ter multiplied  hy  3.14159. 

THEOREM  XVI. 

63*  The  volume  of  a  sphere  is  the  product  of  its  surface  hy  one 
third  of  its  radium. 

A  sphere  may  be  conceived  to  be  composed  of  an  infinite  num- 
ber of  pyramids  whose  vertices  are  at  the  centre  of  the  sphere, 
and  whose  bases,  being  infinitely  small  planes,  coincide  with  the 
surface  of  the  sphere.  The  altitude  of  each  of  these  pyramids 
is  the  radius  of  the  sphere,  and  the  sum  of  the  surfaces  of  their 
bases  is  the  surface  of  the  sphere.  The  volume  of  each  pyra- 
mid is  the  product  of  the  area  of  its  base  by  one  third  of  its 
altitude,  that  is,  of  the  radius  of  the  sphere  (51) ;  and  the  vol- 
ume of  all  the  pyramids,  that  is,  of  the  sphere,  is,  therefore, 
the  product  of  the  surface  of  the  sphere  by  one  third  of  its 
radius. 

64.  Cor.  1.  Let  V  =z  the  volume  of  the  sphere,  and  R,  D, 
and  aS^  the  same  as  in  (62).     Then,  as  (62) 

F=47ri?2  X  \R=z^itR\  or  Jtt/)'' 
That  is,  the  volum.e  of  a  sphere  is  the  cube  of  the  diam,eter  multi- 
plied hy  .5236. 

65.  Cor.  2.  As  in  these  equations  f  it  and  \  it  are  constant, 
the  volumes  of  spheres  vary  as  the  cubes  of  their  radii,  or  as  the 
cubes  of  their  diameters. 


-rf       •  ^ 

BOOK  V.  -  85 

\^)ji  PRACTICAL  QUESTIONS.  ^ 

1.  How  many  square  feet  in  the  convex  -8urf3,c«"of  a  rfght  prism  whose 
altitude  is  2  feet,  and  whose  base  is  o.  regular  hj^agon  of  which  each  side 
is  8  inches  long  ?    How  many  square  feet  in  the  whol^surface  ?     ^^ 

2.  The  radius  of  the  base  of  a  cylinder  is  6  inches,  and  its  altitude  3 
feet ;  how  many  square  feet  in  the  whole  surface  ?    /^~    /  y  / 

3.  What  is  the  number  of  feet  in  the  bounding  planes  of  a  cube  whose 
edge  is  5  feet  ?    The  number  of  solid  feet  in  the  cube  ?    -    /^fiT 

4.  What  is  the  number  of  feet  in  the  bounding  planes  of  a  right  par- 
allelopiped  whose  three  dimensions  are  4,  7,  and  9  feet  ?  The  number  of 
cubic  feet  in  the  parallelopiped  ?,-~-5  A-  \_ 

5.  What  is  the  number  of  cubic  feet  in  the  right  prism  whose  dimen- 
sions are  given  in  the  first  example  ? 

6.  What  is  the  number  of  cubic  feet  in  the  cylinder  whose  dimensions 
are  given  in  the  second  example  ? 

7.  The  altitude  of  a  prism  is  9  feet  and  the  perimeter  of  the  base  6  feet. 
What  is  the  altitude  and  perimeter  of  the  base  of  a  similar  prism  one  third 
as  great  ? 

8.  What  is  the  ratio  of  the  volumes  of  two  cylinders  whose  altitudes  arc 
as  3  :  6,  if  the  cylinders  are  similar  ?  What,  if  the  bases  are  equal  ?  What, 
if  the  bases  are  as  3  :  6  and  the  altitudes  equal  ? 

r"^  How  many  square  feet  in  the  convex  surface  of  a  right  pyramid  whose 
slant  height  is  3  feet,  and  whose  base  is  a  regular  octagon  of  which  each 
side  is  2  feet  long  ? 

10.  How  many  square  feet  in  the  convex  surface  of  a  cone  whose  slant 
height  is  5  feet  and  whose  base  4ias  a  radius  of  2  feel. }  How  many  square 
feet  in  the  whole  surface  ? 

11.  How  many  cubic  feet  in  a  right  quadrangular  pyramid  whose  alti- 
tude is  10  feet,  and  whose  base  is  3  feet  square  ? 

12.  How  many  cubic  feet  in  the  cone  whose  dimensions  are  given  in  the 
tenth  example  ? 

13.  The  slant  height  of  a  frustum  of  a  right  pyramid  is  6  feet,  and  the 
perimeters  of  the  two  bases  are  18  feet  and  12  feet  respectively ;  what  is 
the  convex  surface  of  the  frustum  ? 

14.  What  would  be  the  slant  height  of  the  pyramid  whose  frustum  is 
given  in  the  preceding  example  ? 

15.  What  is  the  whole  surface  of  a  frustum  of  a  cone  whose  altitude  i.s 
8  feet,  and  of  whose  bases  the  radii  are  11  feet  and  5  feet  respectively  ? 


86  SOLID    GEOMETKY. 

10.    The  altitude  of  a  pyramid  is  25  feet,  and  its  base  is  a  rectangle  8 
feet  by  6  ;  how  many  cubic  feet  in  the  pyramid  ? 

17.  The  altitude  of  a  cone  is  20  feet,  and  the  radius  of  its  base  5  feet ; 
how  many  cubic  feet  in  the  cone  ? 

18.  How  many  cubic  feet  in  a  frustum  of  the  cone  given  in  the  preced- 
ing example,  cut  ofi'  by  a  plane  5  feet  from  the  base  ? 

19.  How  far  from  the  base  must  a  cone  whose  altitude  is  12  feet  be  cut 
off  so  that  the  frustum  shall  be  equivalent  to  one  half  of  the  cone  ? 

20.  How  many  square  feet  in  the  surface  of  a  sphere  whose  radius  is  6 
feet  ? 

21.  How  many  cubic  feet  in  a  sphere  whose  radius  is  8  feet  ? 

22.  What  is  the  ratio  of  the  volumes  of  two  spheres  whose  radii  are 
as  4  :  8  y 

23.  Are  spheres  always  similar  solids  ?    Are  cones  ? 

24.  What  is  the  least  number  of  planes  that  can  enclose  a  space  ? 


EXERCISES. 

66.  The  convex  surfaces  of  right  prisms  of  equal  altitudes  are  as 
the  perimeters  of  their  bases.     (14.) 

67  •    The  opposite  faces  of  a  parallelepiped  are  equal  and  parallel. 

68.  The  four  diagonals  of  a  parallelepiped  bisect  each  other. 

69.  A  plane  passing  through  the  opposite  edges  of  a  parallelepiped 
bisects  the  parallelepiped.  • 

70.  In  a  rectangular  parallelepiped  the  diagonals  are  equal ;  and 
the  square  of  each  is  equal  to  the  sum  of  the  squares  of  the  three 
dimensions. 

71.  In  a  cube  the  square  of  a  diagonal  is  three  times  the  squa^re 
of  an  edge. 

72.  Prisms  are  to  each  other  as  the  products  of  their  bases  by 
their  altitudes.     (25.) 

73.  Prisms  with  equivalent  bases  are  as  their  altitudes;  with 
equal  altitudes,  as  their  bases.     (72.) 


BOOK  V.  87 

74.  Polygons  formed  by  parallel  planes  cutting  a  pyramid  are  as 
the  squares  of  their  distances  from  the  vertex,     (39  ;  II.  31.) 

75.  Pyramids  are  to  each  other  as  the  products  of  their  bases  by 
their  altitudes.     (51.) 

76.  Pyramids  with  equivalent  bases  are  as  their  altitudes ;  with 
equal  altitudes,  as  their  bases.     (75.) 

77«    How  can  Theorem  VIII.  be  proved  from  Theorem  IX.  ? 

78.    If  a  pyramid  is  cut  by  a  plane  parallel  to  its  base,  the  pyra* 
mid  cut  off  will  be  similar  to  the  whole  pyramid.     (39 ;  4). 

79«    In  a  sphere  great  circles  bisect  each  other. 

80«    A  great  circle  bisects  a  sphere.     (54.) 

81.  The  centre  of  a  small  circle  is  in  the  perpendicular  from  the 
centre  of  the  sphere  to  the  small  circle. 

82.  Small  circles  equally  distant  from  the  centre  of  a  sphere  are 
equal. 

83.  The  intersection  of  the  surfaces  of  two  spheres  is  the  circum- 
ference of  a  circle. 

84.  The  arc  of  a  great  circle  can  be  made  to  pass  through  any 
two  points  on  the  surface  of  a  sphere.     (IV.  4.) 

85.  Definition.     A  plane  is  tangent  to  a  sphere  when  it  touches 
but  does  not  cut  the  sphere. 

86.  Prove  that  the  radius  of  a  sphere  to  the  point  of  tangency  of 
a  plane  is  perpendicular  to  the  plane.     (IV.  8.) 

87.  As  the  serai-decagon  revolves  about  A  F^  «- 
what  kind  of  a  solid  is  described  by  the  triangle  p 
A  BKl     What  by  the  trapezoid  KC?     By  LD'i  ^ 

88.  The  surface  described  by  the  line  A  B  =  ^ 
AKX  circ.  GO. 

Draw  from  G  a  perpendicular  to  A  B,  and  from 
the  point  where  it  meets  A  B  a.  perpendicular  to     ^ 
A  F.     (42.)  F 

89.  The  surface  described  by  the  line  C D  =  L M  y,  circ.  GO. 
(15.) 


00  SOLID   GEOMETRY. 

90»  Definition.  The  surfaces  described  by  the  arcs  AB,  BC,  CD, 
&c.  are  called  zones. 

91.  The  area  of  a  zone  Is  equal  to  the  product  of  its  altitude  by 
the  circumference  of  a  great  circle. 

92.  Zones  on  the  same  or  equal  spheres  are  as  their  altitudes. 

93.  The  surface  of  a  sphere  is  four  times  the  surface  of  one  of  its 
great  circles.     (62;  III.  32.) 

94.  Definition.  A  polyedron  is  circumscribed  about  a  sphere 
when  its  faces  are  each  tangents  to  the  sphere.  In  this  case  the 
sphere  is  inscribed  in  the  polj^'edron. 

95.  The  surface  of  a  sphere  is  equal  to  the  convex  surface  of  the 
circumscribed  cylinder.     (62  ;  15.) 

96.  Definition.  A  Spherical  Sector  is  the  solid  described  by  any 
sector  of  a  semicircle  as  the  semicircle  revolves  about  its  diameter. 

97.  The  volume  of  a  spherical  sector  is  equal  to  the  product  of  the 
surface  of  the  zone  forming  its  base  by  one  third  of  the  radius  of  the 
sphere  of  which  it  is  a  part. 

98.  A  Spherical  Segment  is  a  part  of  a  sphere  included  by  two 
parallel  planes  cutting  or  touching  the  sphere.  When  one  plane 
touches  and  one  cuts  the  sphere,  the  spherical  segment  is  called  a 
spherical  segment  of  one  base  ;  when  both  cut,  a  spherical  segment  of 
two  bases. 

99.  How  can  the  volume  of  a  spherical  segment  of  one  base  be 
found?    A  spherical  segment  of  two  bases? 

100.  A  sphere  is  two  thirds  of  the  circumscribed  cylinder. 

101.  A  cone,  hemisphere,  and  cylinder  having  equal  bases  and  the 
same  altitude  are  as  the  numbers  1,  2,  3. 


.^ 


BOOK  VI 


PROBLEMS   OF  CONSTRUCTION. 

In  the  preceding  demonstrations  we  have  assumed  that  our 
figures  were  already  constructed.  The  Problems  of  Construc- 
tion given  in  this  Book  depend  for  their  solution  upon  the  prin- 
ciples of  the  preceding  Books.  In  some  of  the  problems  the 
construction  and  demonstration  are  given  in  full ;  in  others  the 
construction  is  given  and  the  propositions  necessary  to  prove 
the  construction  referred  to  in  the  order  in  which  they  are  to 
be  used,  and  the  pupil  must  complete  the  demonstration.  In  a 
few  instances  references  are  made  to  the  Exercises  appended  to 
the  previous  Books.  In  such  cases  either  the  propositions  to 
which  reference  is  made  can  be  demonstrated  or  the  problem 
omitted. 


PROBLEM  1. 


1,    To  bisect  a  given  straight  line. 

Let  ^  ^  be  the  given  straight  line.  From 
A  and  B  as  centres  with  a  radius  greater 
than  half  of  A  B,  describe  arcs  cutting  one 
another  at  C  and  D ;  join  C  and  D  cutting 
AB  hi  U,  and  the  line  ^  -ff  is  bisected  at  U. 
For  C  and  D  being  each  equally  distant  from 
A  and  B,  the  line  CD  must  be  perpendicu- 
lar to  ^  ^  at  its  middle  point  (converse  of 
I.  50). 


C 


D 


B 


90 


PLANE  GEOMETKV. 


PROBLEM   11. 

2.    From  a  given  point  without  a  straight  line  to  draw  a  per- 
pendicular to  that  line. 

Let  G  be  the  point  and  A  B  the  line. 

From  (7  as  a  centre  describe  an  arc 
cutting  ^  ^  in  two  points  E  and  F ;  with       ^ 

E  and  F  as  centres,  with  a  radius  greater    A    >>..^ ■■'p'  ^ 

than  half  E  F,  describe  arcs  intersecting 
at  jD.  Draw  CD,  and  it  is  the  perpen- 
dicular required  (converse  of  I.  53). 


PROBLEM  in. 

3*    From  a  given  point  in  a  straight  line  to  erect  a  perpendicu- 
lar to  that  line. 


Let  C  be  the  given  point  and  A  B  the  F 

given  line.  ^ 

With  (7  as  a  centre  describe  an  arc 
cutting  ^  ^  in  i>  and  E  \  with  B  and  E 
as  centres,  with  a  radius  greater   than    A-\  q  )-B 

D  C,   describe    arcs    intersecting   at  F. 

Draw  C  Fy  and  it  is  the  perpendicular  required  (converse  of 
L  53). 


Second  Method.  With  C  as  a  centre  de- 
scribe an  arc  D  E  F ;  take  the  distances 
BE  and  EF  equal  to  CD,  and  from  E 
and  F  as  centres,  with  a  radius  greater 
than  half  the  distance  from  E  to  F,  de- 
scribe arcs  intersecting  at  G.     Draw  G  G, 


/ 


/ 


z>        c 

and  it  is  the  perpendicular  required  (III.  33 ;  III.  16  -,  III.  15). 


BOOK  VI. 


91 


Third  Method.  With  any  point,  D, 
without  the  line  A  B,  with  a  radius  equal 
to  the  distance  from  D  to  C,  describe  an 
arc  cutting  AB  at  E ;  draw  the  diameter 
ED  F.  Draw  C F,  and  it  is  the  perpen- 
dicular required  (III.  23). 


PROBLEM  IV. 


4t    To  bisect  a  given  arc,  or  angle. 

1st.  Let  AB  he  the  given  arc.  Draw  the 
chord  A  B  and  bisect  it  with  a  perpendicu- 
lar (1;  III.  16). 

2d.    Let  C  be  the  given  angle. 

With  (7  as  a  centre  describe  an  arc  cutting 
the  sides  of  the  angle  in  A  and  B ;  bisect  the 
arc  A  B  with  the  line  CD,  and  it  will  also  bi- 
sect the  angle  C{IU.  11). 


PROBLEM  V. 


5t   At  a  given  point  in  a  straight  line  to  make  an  angle  equal 
to  a  given  angle. 


Let  A  be  the  given  point  in  the  line 
A  B,  and  C  the  given  angle.  With  C 
as  a  centre  describe  an  arc  DF  cut- 
ting the  sides  of  the  angle  C;  with 
^  as  a  centre,  with  the  same  radius, 
describe  an  arc;  with  ii^  as  a  centre, 
with  a  radius  equal  to  the  distance  from 
D  to  E,  describe  an  arc  cutting  the 
arc  FG.     Draw  AG.     The  angle  J  = 


C  (III.   12;  m.   11). 


92 


PLANE  GEOMETRY. 


PROBLEM  VL 

6*    Through  a  given  point  to  draw  a  line  parallel  to  a  given 
straight  line. 


Let  C  be  the  given  point,  and  A  B 
the  given  line.  From  G  draw  a  line 
(7i>  to  ^^;  at  (7  in  the  line  DC 
make  an  angle  D  C  E  equal  to  C  D  A 
(5) ;  CE  is  parallel  to  ^  ^  (I.  18). 


PROBLEM  Vn. 
7»    Two  angles  of  a  triangle  given,  to  find  the  third. 


Draw  an  indefinite  line  AB-,  at 
any  point  C  make  an  angle  AG D 
equal  to  one  of  the  given  angles,  and 
DG E  equal  to  the  other  (5).  Then 
ECB'm  the  third  angle  (I.  7 ;  I.  33). 


PROBLEM  VIII. 

8.    The  three  sides  of  a  triangle  giveriy  to  construct  the  triangle. 

Take  A  B  equal  to  one  of  the  given  sides ; 
with  ^  as  a  centre,  with  a  radius  equal  to 
another  of  the  given  sides,  describe  an  arc, 
and  with  ^  as  a  centre,  with  a  radius  equal 
to  the  remaining  side,  describe  an  arc  inter- 
secting the  first  arc  at  G.  Draw  A  G  and  G B^  and  AGB  ia 
evidently  the  triangle  required. 


BOOK  VI. 


PROBLEM  IX. 


93 


3 


/- 


/   1/ 


9e    Two  sides  and  the  inchided  angle  of  a^trian^e  ^iven,  to 
construct  tke  triangle.     , 

DrS^  A  B  equal  to  one  of  the  given  sides ;  C 

at  B  make  the  angle  ABC  equal  to  the  given 
angle  (5),  and  take  BG  equal  to  the  other 
given  side ;  join  A  and  (7,  and  ABC  m  evi- 
dently the  triangle  required. 


PROBLEM  X. 


10.    Two  angles  and  a  side  of  a  triangle  given,  to  construct 
tlie  triangle. 

If  the  angles  given  are  not  both  adja- 
cent to  the  given  side,  find  the  third  angle 
by  (7).  Then  draw  A  B  equal  to  the  given 
side,  and  at  B  make  an  angle  ABC  equal 
to  one  of  the  angles  adjacent  to  A  B,  and 
at  A  make  an  angle  B  AC  equal  to  the  other  angle  adjacent  to 
A B,  and  A  BC  i^  evidently  the  triangle  required. 


PROBLEM  XL 


1 1 ,    Two  sides  of  a  triangle  and  the  angle  opposite  one  of  them 
given,  to  construct  the  triangle. 

Draw  an  indefinite  line  AC ;  at  A 
make  the   angle   CAB   equal  to   the 
given  angle,  and  take  A  B  equal  to  the      .  j 
side  adjacent  to  the  given  angle ;  with 
B  as  a  centre,  with  a  radius  equal  to  the  other  given  side,  de- 
scribe an  arc  cutting  A  C.     If  the  given  angle  A  is  acute, 


D- 


^G 


94  PLANE   GEOMETRY. 

1st.  The  given  side  B  C,  opposite  the  given  angle,  may  be 
less  than  the  other  given  side  ;   then  -^ 

the  arc  described  from  B  as  a  centre 
will  cut  AC  in  two  points,  C  and  D, 

on  the  same  side   of  A,  and,  drawing  ^^-^A,---' 

BC  and  BD,  the  triangles  ABC  and  ABD  (whose  angle  BDA 
is  the  supplement  of  the  angle  BCA),  both  satisfy  the  given 
conditions. 

2d.  The  given  side  opposite  the  given  angle  may  be  equal  to 
the  perpendicular  B  E ;  then  the  arc  described  from  i?  as  a 
centre  will  tonch  A  (7,  and  the  right  triangle  AB E  h  the  only 
one  that  can  satisfy  the  given  conditions. 

3d.  The  side  opposite  the  given  angle  may  be  greater  than 
the  other  given  side  ;  then  the  arc  described  from  ^  as  a  centre 
will  cut  A  C  in  C,  and  in  another  point  on  the  other  side  of  A. 
In  this  case  there  can  be  but  one  triangle  ABC  satisfying  the 
given  conditions,  the  triangle  formed  on  the  opposite  side  of 
A  B  containing  not  the  given  angle  but  its  supplement. 

4th.  If  the  given  angle  is  not  acute,  the  given  side  opposite 
the  given  angle  must  be  greater  than  the  other  given  side,  and, 
as  in  the  last  case  above,  there  can  be  but  one  solution. 

12*  Scholium,  If  the  side  opposite  the  given  angle  A  is 
less  than  the  perpendicular,  or  if  the  given  angle  is  not  acute, 
and  at  the  same  time  the  side  opposite  the  given  angle  is  less 
than  the  other  given  side,  the  solution  is  impossible. 

13.  Corollary.  From  this  and  the  preceding  Problem  and 
Theorems  VIII.,  IX.,  and  XIV.  of  Book  I.,  it  follows  that  witli 
the  exception  of  the  ambiguity  pointed  out  in  the  first  part  of 
this  Problem,  two  triangles  are  equal  if  any  three  parts,  of  which 
one  is  a  side,  of  the  one  are  equal  to  the  corresponding  parts  of 
the  other. 


BOOK   VI. 


95 


PROBLEM   XII. 

14*  To  find  the  centre  of  a  given  circumference  or  of  a  given 
arc. 

Let  ABB  he  the  given  circumference, 
or  arc. 

Draw  any  two  chords  not  parallel  to 
each  other,  a,s  A  B,  B  D,  and  bisect  these 
chords  by  the  perpendiculars  C  B  and 
CF.  These  perpendiculars  will  intersect 
at  the  centre  of  the  circumference  or 
arc  (III.  17). 

15 1  Scholium.  By  the  same  construction  a  circumference 
may  be  made  to  pass  through  any  three  given  points ;  or  a  cir- 
cle circumscribed  about  a  given  triangle ;  or  about  a  given 
regular  polygon  (II.  34). 

PROBLEM  XIIL 
16t    To  inscribe  a  circle  in  a  given  triangle. 

Let  ABChe  the  given  triangle. 

Bisect  any  two  of  its  angles,  as  A  and  G, 
D^  where  the  two  bisecting  lines  meet,  as 
a  centre,  with  a  radius  equal  to  the  distance 
of  D  from  any  one  of  the  sides,  describe  a 
circle,  and  it  will  be  the  circle  required. 

Draw  the  perpendicular  DE,  DF,  DG. 
The  angles  at  A  are  equal  by  construction, 
and  the  angles  AED  and  AFD  are  each 
right  angles ;  therefore  the  triangles  AD  E 
and  AFD  are  mutually  equiangular  (I.  35), 
and  the  hypothenuse  A  D  m  common;  therefore  the  triangles  are 
equal  (I.  41),  and  D  E  =  D  F.  In  like  manner  D  E  =  D  G. 
Therefore  the  circle  described  from  />  as  a  centre  with  the  radius 
D  E  will  pass  through  the  points  7^  and  G ;  and  since  the  angles 
at  Ef  F,  G  are  right  angles,  the  sides  of  the  triangle  A  BG  are 


With  the  point 
A 


96 


PLANE  GEOMETRY. 


tangents;   therefore  the  circle  EFG  m  inscribed  in  the  tri- 
angle ABC  (III.  20). 

17.   Scholium.     The  lines  bisecting  the  angles  of  a  triangle 
all  meet  in  the  same  point. 


PROBLEM  XIV. 

18.    Through  a  given  point  to  draw  a  tangent  to  a  given  cir- 
cumference. 

1st.    If  the  given  point  is  in  the  circumference. 

Erect  a  perpendicular  to  the  radius  at  the  given  point  (III.  47). 

2d.    If  the  given  point  is  without 
the  circumference. 

Join  the  given  point  A  with  the 
centre  C  of  the  given  circle  B  D  E ; 
on  ^  C  as  a  diameter  describe  a  cir- 
cle cutting  the  given  circle  in  B  and 
D.  Draw  A  B  and  A  B,  and  each  will  be  tangent  to  the  given 
circle  through  the  given  point.  For  drawing  the  radii  CB,  CD, 
the  angles  .5,  D  are  each  right  angles  (III.  23) ;  therefore  A  B, 
A  B  are  tangents  to  the  given  circle. 

19i    Corollary.     The  tangents  A  B,  AD  are  equal  (I.  50). 


PROBLEM   XV. 

20*    Upon  a  given  straight  line  to  describe  a  segment  of  a  circle 
which  shall  contain  a  given  angle. 

Let  ^  ^  be  the  given  straight  line. 

At  B  make  the  angle  ABB  equal  to 
the  given  angle  (5).  Draw  B  C  perpen- 
dicular to  D  B ;  bisect  A  B  in  B,  and 
from  B  draw  EC  perpendicular  to  A  B. 
From  C,  the  point  of  intersection  of 
BC  and  EC,  with  a  radius  equal  to 
G By  describe  a  circle  AG B F ^  B FA  is  the  segment  required. 


BOOK   VI.  97 

A  B  is  ?i  chord  (I.  53).  And  as  ^  Z)  is  perpendicular  to  the 
radius  C  B  at  B,  B  D  is  a  tangent  to  the  circle,  and  hence  the 
angle  A  B  I)  m  measured  by  half  the  arc  AG  B  (III.  54) ;  and 
any  angle  B  F  A  inscribed  in  the  segment  B  F  A  is  ako  meas- 
ured by  half  the  arc  AG B  (III.  21),  and  is  therefore  equal  to 
the  angle  A  B  D  or  the  given  angle. 

21*  Corollary.  If  the  given  angle  is  a  right  angle,  the  re- 
quired segment  would  be  a  semicircle  described  on  the  given 
line  as  a  diameter. 


PROBLEM  XVI. 
22t    To  divide  a  given  line  into  parts  proportional  to  given  lines. 

Let  it  be  required  to  divide    A 
A  B    into   parts   proportional 
to  Jf,  N,  0. 

Draw  at  any  angle  with 
^  ^  an  indefinite  line  A  C. 
From  A  cut  o^  A  D,  D  E,  E  F  equal  respectively  to  i¥,  N,  0. 
Join  B  to  F,  and  through  D  and  E  draw  lines  parallel  to  B  F. 
These  parallels  divide  the  line  as  required  (II.  16). 

23.    Corollary.    By  taking  J/,  N,  0  equal,  the  given  line  can 
be  divided  into  equal  parts. 


PROBLEM  XVII. 
24.    To  find  a  fourth  proportional  to  three  given  lines. 

Let  it  be  required  to  find 
a  fourth  proportional  to  M, 
N,0.^  A 

Draw  at  any  angle  with 
each  other  the  indefinite 
lines  AF,  AG.     From  AF  cut  off  AB=zM,  BC  =  y,  and 


98 


PLANE  GEOMETRY. 


from  A  G  cut  off  A  I)  =z  0.  Join  B  D  and  through  G  draw 
G  E  parallel  to  B  D  \  then  D  E  i&  the  required  fourth  pro- 
portional (II.  16). 

25%  Gorollary.  By  taking  A  B  equal  to  Jf,  and  A  D  and 
B  G  each  equal  to  N^  a  third  proportional  can  be  found  to  M 
and  N. 


PROBLEM  XVIII. 
26  •    To  find  a  mean  proportional  between  two  given  lines. 

Let  it  be  required  to  find  a  mean     M  N 
proportional  between  M  and  N. 

From  an  indefinite  line   cut  off" 
AB  =  M,  BG  =  J^;  on  AG  as  a 
diameter  describe  a  semicircle,  and 
at  B  draw  B  D  perpendicular  to  A  G. 
tional  required.     Join  AD,  D  G. 


BB  m  the  mean  propor- 
(III.  23 ;  11.  26.) 


27.  Definition.  When  a  line  is  divided  so  that  one  segment 
is  a  mean  proportional  between  the  whole  line  and  the  other 
segment,  it  is  said  to  be  divided  in  extreme  and  mean  ratio. 


PROBLEM  XIX. 
28.    To  divide  a  given  line  in  extreme  and  mean  ratio. 

Let  it  be  required  to  divide  A  B  m  extreme  and  mean  ratio. 

At  B  draw  the  perpendicular 
BG  =  \  AB;  join  AG',  cut  off 
G  D  =  G  B,  A  E  =z  A  D,  QXidi  A  B  m 
divided  at  E  in  extreme  and  mean 
ratio. 

For,  describe  a  circle  with  the  cen- 
tre G  and  radius  G  B  and  produce  A  G  to  meet  the  circumfer- 
ence in  Ft  then  ^1  i^  is  a  secant  and  AB  ^  tangent  of  the  circle 
DFB,  and  therefore  (III.  64) 


BOOK  VI. 
AF'.AB  =  AB:AD 


99 


and  (Pn.  18) 

AF—AB'.ABz=AB  —  AD:AD 
But  AB=2CB=:DF 

therefore  AF— AB  =  AF— DF=AD  =  AE 

and  the  proportion  becomes 

AF  :AB  =  FB  :AF 
or  (Pn.  16)  AB:AF  =  AF:FB 

PROBLEM  XX. 

29.  Through  a  given  point  ivithin  the  sides  of  a  given  angle  to 
draw  a  line  so  that  the  segments  included  between  the  j^oint  and 
the  sides  of  the  angle  may  he  in  a  given  ratio. 


MN 


Let  it  be  required  to  draw  through 
the  point  D  within  the  angle  ^  a  line 
so  that  AD  '.DC  =^M  :N. 

Draw  D  E  parallel  to  A  B. 

Find  EC  2,  fourth  proportional  to 
M,  N,  and  BE  (24);  join  C  to  D, 
and  produce  CD  to  J,  and  AC  \^  the  line  required  (XL  16). 


PROBLEM  XXI. 

30*    The  base,  an  adjacent  angle,  and  the  altitude  of  a  triaiufle 
given,  to  construct  the  triangle. 

At  A  of  the  base  A  B  draw  an  indefi- 
nite line  A  C  making  the  angle  A  equal 
to  the  given  angle ;  at  any  point  in  A  B, 
as  D,  draw  the  perpendicular  DE  equal  ^  ^ 

to  the  given  altitude  ;  through  E  draw  EF  parallel  to  ^  J5  cut- 
ting AC  in  G',  join  G  j5,  and  A  G  B  'i&  the  triangle  required. 


100  PLANE  GEOMETRY. 

PROBLEM  XXIL 

31 «  To  construct  a  parallelogram,  having  the  sum  of  its  base 
and  altitude  given,  tohick  shall  be  equivalent  to  a  given  square. 

On  A  B,  the  given  sum,  as  a  diame- 
ter, describe  a  semicircumference.  At 
any  point,  as  B,  \n  A  B  draw  the  perpen- 
dicular B  C  equal  to  a  side  of  the  given 
square ;  through  C  draw  C  D  parallel  to 
A  B,  cutting  the  circumference  in  D  ;  draw  D  E  perpendicular 
io  AB.  A  E,  E  B  are  one  the  base  and  the  other  the  altitude 
of  the  parallelogram  required  (26). 

32.  Scholium.  If  the  side  of  the  square  is  greater  than 
half  the  sum  of  the  base  and  altitude,  the  construction  is  im- 
possible. 

PROBLEM  XXIII. 

33*  To  construct  a  parallelogram  having  the  difference  between 
its  base  and  altitude  given,  which  shall  be  equivalent  to  a  given 
square. 

On  A  B  the  given  difference,  as  a  diameter, 
describe  a  ch'cumference.  At  A  draw 
the  perpendicular  A  Z)' equal  to  a  side  of  the 
given  square ;  join  D  with  the  centre  C,  and 
produce  D  C  to  E.  D  F,  D  E  are  one  the 
base  and  the  other  the  altitude  of  the  paral- 
lelogram required  (III.  64). 


PROBLEM  XXIV. 

34*    To  construct  a  square  equivalent  to  a  given  parallelogram. 

Find  a  mean  proportional  between  tjie  altitude  and  base  of 
the  given  parallelogram  (26),  and  it  will  be  a  side  of  the  re- 
quired square. 


Q^^y^- 


BOOK  Nl.  101 

PROBLEM  XXV. 

35*    To  construct  a  square  equivalent  to  a  given  triangle. 

Find  a  mean  proportional  between  the  base  and  half  the 
altitude  (26),  and  it  will  be  a  side  of  the  required  square. 

PROBLEM  XXVI. 

36.^  To  construct  a  square  equivalent  to  a  given  circle. 

Find  a  mean  proportional  between  the  radius  and  the  semi- 
circimiference,  and  it  will  be  a  side  of  the  required  square. 

PROBLEM  XXVII. 

37.  To  construct  a  square  equivalent  to  the  sum  of  two  given 
squares. 

Construct  a  right  triangle  (9)  with  the  sides  adjacent  to  the 
right  angle  equal  respectively  to  the  sides  of  the  given  squares ; 
the  hypothenuse  will  be  a  side  of  the  required  square  (II.  27). 

38.  Scliolium.  By  continuing  the  same  process  we  can  find 
a  square  equivalent  to  the  sum  of  any  number  of  given  squares. 

PROBLEM  XXVIII. 

39.  To  construct  a  square  equivalent  to  the  difference  of  two 
given  squares. 

Construct  a  right  .triangle  (11),  taking  as  the  hypothenuse  a 
side  of  the  greater  square,  and  for  one  of  the  sides  adjacent  to 
the  right  angle  a  side  of  the  other  square ;  the  third  side  of 
the  triangle  will  be  a  side  of  the  required  square  (II.  28). 


102 


PLANE  GEOMETRY. 


PROBLEM   XXIX. 


E  D  F 

For  the  triangles  BCD 


40i    To  construct  a  triangle  equivalent  to  a  given  polygon. 

Let  A  Dhe  the  polygon.  ^ 

Draw  B  D  cutting  off  the  triangle 
^C  D\^xk^\\^  Q  draw  CF  parallel 
to  ^  Z>  meeting  £  1)  produced  in  F; 
JQia  ^  F,  .afn.d  a.  polygon  A  B  F  E  is 
formed  with  one  side  less  than  the 
given  polygon  and  equivalent  to  it. 
and  B  F  D,  having  the  same  base  B  D,  and  the  same  altitude, 
are  equivalent ;  adding  to  each  the  common  part  A  B  D  E,  we 
have  ABODE  equivalent  to  A  B  F  E.  In  like  manner  a 
polygon  with  one  side  less  can  be  found  equivalent  to  A  B  F  E, 
and  by  continuing  the  process  the  sides  may  be  reduced  to  three, 
and  a  triangle  obtained  equivalent  to  the  given  polygon. 

41.  Scholium.  Since  by  (35)  a  square  can  be  found  equiva- 
lent to  a  given  triangle,  by  (40)  and  (35)  a  square  can  be  found 
equivalent  to  any  polygon. 


PROBLEM  XXX. 


42«    On  a  given  line  to  construct  a  polygon  similar  to  a  given 
polygon. 

Let  AD  \)Q  the  given  poly- 
gon and  JfZ  the  given  line. 

Draw  the  diagonals  A  E^ 
AD,  AO.  At  M  and  L 
make  the  angles  G ML  and         F  E 

G LM  equal  respectively  to  A  FE  and  A  E F,  and  a  triangle 
G  L  M  will  be  formed  similar  to  A  E  F.  In  like  manner  on 
G  L  construct  a  triangle  similar  to  AD  E)  on  G  K  one  similar 
to  AO  D ;  on  GI  one  similar  to  ABO ;  and  the  polygons  A  D, 


BOOK  VI. 


103 


KG,  being  composed  of  the  same  number  of  similar  triangles 
similarly  situated,  are  similar  (11.  75), 


PROBLEM  XXXI. 

43*  Tim  similar  polygons  being  given,  to  construct  a  similar 
polygon  equivalent  to  their  sum,  or  to  their  difference. 

Find  a  line  whose  square  shall  be  equivalent  to  the  sum  (37), 
or  to  the  difterence  (39),  of  the  squares  of  any  two  homologous 
sides  of  the  given  polygons,  and  this  will  be  the  homologous 
side  of  the  required  polygon  (11.  31).  On  this  line  construct 
(42)  a  polygon  similar  to  the  given  polygons. 


PROBLEM  XXXIL 

44 «    To  construct  a  square  which  shall  he  to  a  given  squa^'e  in  a 
given  ratio. 

On  any  line  ^  (7,  as  a  diameter, 
describe  a  semicircumference  ABC; 
divide  the  line  A  C  at  the  point  D 
so  that  A  D  :  D  C  in  the  given  ra- 
tio. Perpendicular  to  AC  draw  D  B 
meeting  the  circumference  at  B ;  join  B  A,  B  C,  and  on  BC, 
produced  if  necessary,  take  BF=  a,  side  of  the  given  square. 
Through  F  draw  JE  F  parallel  to  A  C,  meeting  B  A  in  E,  and 
B  E  is  &  side  of  the  required  square. 

For  as  -5  is  a  right  angle  (TIL  23),  we  have  (IL  .72) 
BF^  .BF^  —  EG  :  GF 

But  as  ^i^  is  parallel  to  A  C,  we  have  (II.  47) 
EG  .GF=AD  '.DC 
therefore  (Pn.  11) 

BE^'.BF^=zAD  .DC 


104 


PLANE  GEOMETRY. 


PROBLEM  XXXIII. 

45*    To  inscribe  a  square  in  a  given  circle. 

Draw  two  diameters  AC,  B  D  2!  right 
angles  to  each  other,  and  join  A  B, 
B C,  CD,  DA]  ABCn  is  the  required 
square  (III.  23 ;  III.  12). 

46*  Corollary.  By  bisecting  the  arcs 
AB,  BC,  CD,  DA,  and  drawing  the 
chords  of  these  smaller  arcs,  a  regular 
octagon  will  be  inscribed  in  the  circle.  By  continuing  this 
bisection  regular  polygons  can  be  inscribed  having  the  number 
of  their  sides  16,  32,  64,  and  so  on. 


PROBLEM  XXXIY. 

47t    To  inscribe  a  regular  hexagon  in  a  given  circle. 

TaKe  A  B  equal  to  the  radius  of  the 
given  circle,  and  it  will  be  a  side  of  the 
hexagon  required  (III.  33). 

48.  Corollary.  By  drawing  A  C,  CD, 
D  A  2in  equilateral  triangle  will  be  in- 
scribed in  the  circle.  By  bisecting  the 
arcs  AB,  BC,  &c.,  and  continuing  this 
bisection  as  in  (46),  and  drawing  the  chords  of  these  smaller 
arcs,  regular  polygons  can  be  inscribed  having  the  number  of 
their  sides  12,  24,  48,  96,  and  so  on. 


PROBLEM  XXXV. 


49.    To  inscribe  a  regular  decagon  in  a  given  circle. 

Divide  the  radius  ^  ^  in  extreme  and  mean  ratio  at  the  point 
D  (28),  and  take  BC  =  A  D,  the  greater  segment,  and  it  will 
be  the  side  of  the  required  decagon. 


BOOK  VI. 


105 


Draw  AC,  CJD.  The  triangles  ACB, 
DCB  are  similar  (II.  23);  for  they  have 
the  angle  B  common,  and  by  construction 

AB  :AD=iAD  :  D  B 
but  AD  =  BG 

therefore    AB:BC  =  BC:BD  B        C 

Therefore,  as  A  C  B  is  isosceles,  I)  C  B  is  also  isosceles,  and 
CD  =  C B ;  therefore  also  C  J)  =  I)  A,  and  A  C D  is  an  isos^ 
celes  triangle,  and  the  angle  A  =  A  C  B.  But  the  exterior- 
angle  BI)C  =  A-\-ACI)  =  twice  the  angle  A.  Therefore, 
as  ^  =  BB  C,  B  =  twice  the  angle  A.  But  B  =  AC  B; 
therefore  the  sum  of  the  three  angles  A,  B,  and  A  C B  is  equal 
to  five  times  the  angle  A  ;  or  the  angle  A  is  one  fifth  of  two 
right  angles,  or  one  tenth  of  four  right  angles;  therefore  the 
arc  B  C  is  one  tenth  of  the  circumference,  and  the  chord  B  C 
a  side  of  a  regular  decagon  inscribed  in  the  circle. 

50«  Corollary.  By  drawing  chords  joining  the  alternate 
vertices  a  regular  pentagon  will  be  inscribed.  By  proceeding  as 
in  (46)  regular  polygons  can  be  inscribed  having  the  number  of 
their  sides  20,  40,  80,  and  so  on. 


PROBLEM  XXXVI. 

51.  To  inscribe  a  regular  polygon  of  fifteen  sides  in  a  given 
circle. 

Find  by  (47)  the  arc  A  C  equal  to  a  sixth 
of  the  circumference,  and  by  (49)  the  arc 
A  B  equal  to  a  tenth  of  the  circumference, 
and  the  chord  B  C  will  be  a  side  of  the  poly- 
gon required. 

For  i  -  T^  =  -iij 

52.  Corollary.     Proceeding  as  in  (46)  regular  polygons  can 
be  inscribed  having  the  number  of  their  sides  30,  60,  and  so  on. 


106  PLANE  GEOMETRY. 

PROBLEM  XXXVIL 

53*  To  circumscribe  about  a  given  circle  a  polygon  similar  to  a 
given  inscribed  regular  polygon. 

Let  A  D  hQ  the  given  inscribed  poly- 
gon.    Through  the  points  A,  B,  C,  B, 
E,  F  draw  tangents  to  the  circumference.  M^ 
These  tangents  intersecting  will  form  the 
polygon  required. 

For  the  triangles  AGB,  B  H  0,  &c. 
are  isosceles  (19);  and  as  the  arcs  AB, 
B  (7,  &c.  are  equal,  the  angles  GAB, 
GBA,  HBC,  RGB,  &c.  are  equal 
(III.  54) ;  therefore,  as  the  bases  AB,  BG,  &c.  are  equal,  these 
isosceles  triangles  are  equal.  Hence  the  angles  G^  H,  I,  Ky 
L,  M  are  equal,  and  the  polygon  MI  is  equiangular;  and  as 
GB=zBH=HC  =  GI,  &c.,  GH=HI,ko.',  therefore  the 
polygon  MI  is  equilateral  and  regular  (11.  32).  It  is  also  sim- 
ilar to  ^  i>  (II.  33) ;  and  as  its  sides  are  tangents  it  is  circum- 
scribed about  the  circle. 

54 •  Corollary.  As  (45-52)  regular  polygons  can  be  in- 
scribed having  the  number  of  their  sides  3,  4,  5,  6,  8,  10,  12, 
15,  16,  20,  24,  30,  32,  40,  48,  60,  64,  80,  96,  and  so  on,  regu- 
lar polygons  having  the  number  of  their  sides  represented  by 
these  numbers  can  also  be  circumscribed  about  a  given  circle. 

EXERCISES. 

5^»  From  two  given  points  to  draw  two  equal  lines  meeting  in  a 
given  straight  line.     (I.  53.) 

56.  Through  a  given  point  to  draw  a  line  at  equal  distances  from 
two  other  given  points. 

57.  From  a  given  point  out  of  a  straight  line  to  draw  a  line  mak- 
ing a  given  angle  with  that  Hne.     (I.  17.) 


BOOK  VI.  107 

58.  From  two  given  points  on  the  same  side  of  a  given  line  to 
draw  two  lines  meeting  in  the  first  line  and  making  equal  angles 
with  it. 

59.  From  a  given  point  to  draw  a  line  making  equal  angles  with 
the  sides  of  a  given  angle. 

60.  Through  a  given  point  to  draw  a  line  so  that  the  parts  of  the 
line  intercepted  between  this  point  and  perpendiculars  from  two 
other  given  points  shall  be  equal. 

If  the  three  points  are  in  a  straight  line,  the  parts  equal  what  ? 

61 .  From  a  point  without  two  given  lines  to  draw  a  line  such 
that  the  part  between  the  two  lines  shall  be  equal  to  the  part  between 
the  given  point  and  the  nearer  line. 

When  is  the  Problem  impossible  ? 

62.  To  trisect  a  right  angle. 

63.  On  a  given  base  to  construct  an  isosceles  triangle  having 
each  of  the  angles  at  the  base  double  the  third  angle. 

-4-64.    To  construct  an  isosceles  triangle  when  there  are  given 
1st.   The  base  and  opposite  angle. 
2d.   The  base  and  an  adjacent  angle. 
3d.   A  side  and  an  opposite  angle. 
4th.   A  side  and  the  angle  opposite  the  base. 

65.  The  base,  opposite  angle,  and  the  altitude  given,  to  construct 
the  triangle.     (III.  22.)     (20.) 

When  is  the  Problem  impossible  ? 

66.  The  base,  an  angle  at  the  base,  and  the  sum  of  the  sides  given, 
to  construct  the  triangle. 

When  is  the  Problem  impossible  ? 

67.  The  base,  an  angle  at  the  base,  and  the  diflference  of  the  sides 
given,  to  construct  the  triangle. 

1st.    When  the  given  angle  is  adjacent  to  the  shorter  side. 
2d.    When  the  given  angle  is  adjacent  to  the  longer  side. 
When  is  the  Problem  impossible  ? 

68.  The  base,  the  difference  of  the  sides,  and  the  difference  of 
the  angles  at  the  base  given,  to  construct  the  triangle. 


108  PLANE  GEOMETRY. 

69«    The  base,  the  angle  at  the  vertex,  and  the  sum  of  the  sides 
given,  to  construct  the  triangle. 
When  is  the  Problem  impossible  ? 

70»  The  base,  the  angle  at  the  vertex,  and  the  difference  of  the 
sides  given,  to  construct  the  triangle. 

71  •  On  a  given  base  to  construct  a  triangle  equivalent  to  a  given 
triangle. 

72.  With  a  given  altitude  to  construct  a  triangle  equivalent  to  a 
given  triangle. 

73.  Two  sides  of  a  triangle  and  the  perpendicular  to  one  of  them 
from  the  opposite  vertex  given,  to  construct  the  triangle. 

74.  Two  of  the  perpendiculars  from  the  vertices  to  the  opposite 
sides  and  a  side  given,  to  construct  the  triangle. 

1st.   When  one  of  the  perpendiculars  falls  on  the  given  side. 
2d.   When  neither  of  the  perpendiculars  falls  on  the  given  side. 

75.  An  angle  and  two  of  the  perpendiculars  from  the  vertices  to 
the  opposite  sides  given,  to  construct  the  triangle. 

1st.  When  one  of  the  perpendiculars  falls  from  the  vertex  of  the 
given  angle. 

2d.  When  neither  of  the  perpendiculars  falls  from  the  vertex  of 
the  given  angle. 

76.  An  angle  and  the  segments  of  the  opposite  side  made  by  a 
I  perpendicular  from  the  vertex  given,  to  construct  the  triangle. 

It*    G-iven  an  angle,  the  opposite  side,  and  the  line  from  the  given 
-wertex  to  the  middle  of  the  given  side,  to  construct  the  triangle. 
When  is  the  Problem  impossible  ? 

78.  An  angle,  a  perpendicular  from  another  angle  to  the  opposite 
side,  and  the  radius  of  the  circumscribed  circle  given,  to  construct  the 
triangle. 

When  is  the  Problem  impossible  ? 

79.  To  divide  a  triangle  into  two  parts  in  a  given  ratio, 
1st.   By  a  line  drawn  from  a  given  point  in  one  of  its  sides. 
2d.  By  a  line  parallel  to  the  base. 


,v^^' 


yyyvryvt/^^ 


BOOK  VI.  109 


80«/To'*t!iksect  a  triangle  by  straight  lines  drawn  from  a  point 
witbm  to  the  vertices. 

^  /  81«    Parallel  to  the  base  of  a  triangle  to  draw  a  line  equal  to  the 
sum  of  the  lower  segmentg  oLtbe  two  sides. 

82.  Parallel  to  the  base  of  a  triangle  to  draw  a  line  equal  to  the 
diflference  of  the  lower  segments  of  the  two  sides. 

83.  To  inscribe  in  a  given  triangle  a  quadrilateral  similar  to  a 
given  quadrilateral. 

84.  To  divide  a  given  line  so  that  the  sum  of  the  squares  of  the 
parts  shall  be  equivalent  to  a  given  square. 


V85 


t    To  construct  a  parallelogram  when  there  are  given, 
1st.   Two  adjacent  sides  and  a  diagonal. 


2d.   A  side  and  two  diagonals. 


3d.   The  two  diagonals  and  the  angle  between  them. 
4th.   The  perimeter,  a  side,  and  an  angle. 

86i    To  construct  a  square  when  the  diagonal  is  given. 

87 •    To  construct  a  parallelogram  equivalent  to  a  given  triangle 
and  having  a  given  angle. 

88.  To  draw  a  quadrilateral,  the  order  and  magnitude  of  all  the 
ddes  and  one  angle  given. 

Show  that  sometimes  there  may  be  two  different  polygons  satisfy- 
ing the  conditions. 

89.  To  draw  a  quadrilateral,  the  order  and  magnitude  of  three 
sides  and  two  angles  given. 

1st.   The  given  angles  included  by  the  given  sides. 

2d.    The  two  angles  adjacent,  and  one  adjacent  to  the  unknown  side. 

3d.   The  two  angles  being  opposite  each  other. 

4th.   The  two  angles  being  both  adjacent  to  the  unknown  side. 

In  any  of  these  cases  can  more  than  one  quadrilateral  be  drawn  ? 

90.  To  draw  a  quadrilateral,  the  order  and  magnitude  of  two 
sides  and  three  angles  given. 

1st.   The  given  sides  being  adjacent. 
2d.   The  given  sides  not  being  adjacent. 


110  PLANE   GEOMETRY. 

91  •  In  a  given  circle  to  inscribe  a  triangle  similar  to  a  given 
triangle. 

92.  Through  a  given  point  to  draw  to  a  given  circle  a  secant  such 
that  the  part  within  the  circle  may  be  equal  to  a  given  line. 

93 •    With  a  given  radius  to  draw  a  circumference, 
•  1st.   Through  two  given  points. 

2d.   Through  a  given  point  and  tangent  to  a  given  line. 
3d.   Through  a  given  point  and  tangent  to  a  given  circumference. 
4th.    Tangent  to  two  given  straight  lines. 

5th.   Tangent  to  a  given  straight  line  and  to  a  given  circumference. 
6th.   Tangent  to  two  given  circumferences. 

State  in  each  of  these  cases  how  many  circles  can  be  drawn,  and 
when  the  construction  is  impossible. 

94  •    To  draw  a  circumference, 

1st.   Through  two  given  points  and  with  its  centre  in  a  given  line. 

2d.  Through  a  given  point  and  tangent  to  a  given  line  at  a  given 
point. 

3d.  Tangent  to  a  given  line  at  a  given  point,  a,nd  also  tangent  to  a 
second  given  line. 

4th.   Tangent  to  three  given  lines. 

5th.    Through  two  given  points  and  tangent  to  a  given  line. 

6th.   Through  a  given  point  and  tangent  to  two  given  lines. 

95*    To  draw  a  tangent  to  two  circumferences. 

There  can  be  drawn, 

1st.   When  the  circles  are  external  to  each  other,  four  tangents. 

2d.   When  the  circles  touch  externally,  three. 

3d.   When  the  circles  cut,  two. 

4th.   When  the  circles  touch  internally,  one. 

6th.   When  one  circle  is  within  the  other,  none. 


PLANE  TRIGONOMETKY. 


Af 


4:jhapter  i. 


^PKELIMINAEY. 

C     ^  LOGARITHMS. 

1(  Logarithms  are  exponents  of  the  powers  of  some  number 
which  is  taken  as  a  base.  In  the  tables  of  Logarithms  in  common 
use,  the  number  10  is  taken  as  the  base,  and  all  numbers  are  con- 
sidered as  powers  of  10. 

And,  since 

10*^  =  1,    that  is,  since  the  Logarithm  of    1  is  0, 
10^  =  10,         "  "  "  10  "  1, 

102=:  100,       "  "  "  100  "  2, 

10»=1000,     "  «  "         1000  "  3, 

&c.,  <fec.,  &c., 

the  Logarithm  of  any  number  between  1  and  10  is  between 
0  and  1,  that  is,  is  a  fraction;  the  Logarithm  of  any  number 
between  10  and  100  is  between  1  and  2,  that  is,  is  1  plus  a 
fraction ;  and  the  Logarithm  of  any  number  between  100  and 
1000  is  2  plus  a  fraction ;  and  so  on. 

And,  as 

10"^  =  0.1,     that  is,  since  the  Logarithm  of  0.1      is  — 1, 

10-2  =  0.01,         "  «  «  0.01    «— 2, 

10-»  =  0.001,       "  «  "  O.OOI"— 3, 

&c.,  <kc.,  &o., 

the  Logarithm  of  any  number  between  1  and  0.1  is  between 
0  and  — 1,  that  is,  is  — 1  plus  a  fraction ;  the  Logarithm  of  any 

1  A 


2  PLANE  TRIGONOMETRY. 

nuruber  between  0.1  and  0.01  is  between  — 1  and  — 2,  that  is, 
is  — 2  plus  a  fraction;  and  so  on.  The  Logarithms  of  most 
numbers,  therefore,  consist  of  an  integer,  either  positive  or 
negative,  and  a  fraction,  which  is  always  positive.  The  repre- 
sentation of  the  Logarithms  of  all  numbers  less  than  a  unit  by 
a  negative  integer  and  a  positive  fraction  is  merely  a  matter 
of  convenience. 

2i  The  integral  part  of  a  Logarithm  is  called  the  character- 
istic, and  is  not  generally  written  in  the  tables,  but  can  be 
found  by  the  following 

Rule. 

The  characteristic  of  the  Logarithm  of  any  number  is  equal  to 
the  number  of  places  by  which  its  first  significant  figure  on  the  left 
is  removed  from  units'  place,  the  characteristic  being  positive  when 
this  figure  is  to  the  left  and  negative  when  it  is  to  the  right  of 
units'  place. 

E.  g.  The  Logarithm  of  59  is  1  plus  a  fraction ;  that  is,  the 
characteristic  of  the  Logarithm  of  59  is  1.  The  Logarithm  of 
5417.7  is  3  plus  a  fraction;  that  is,  the  characteristic  of  the 
Logarithm  of  5417.7  is  3.  The  Logarithm  of  0.3  is  — 1  plus 
a  fraction;  that  is,  the  characteristic  of  the  Logarithm  of  0.3 
is  — 1.  The  Logarithm  of  0.00017  is  — 4  plus  a  fraction ;  that 
is,  the  characteristic  of  the  Logarithm  of  0.00017  is  — 4. 

3*  Since  the  base  of  this  system  of  Logarithms  is  10,  if  any 
number  is  multiplied  by  10,  its  Logarithm  will  be  increased 
by  a  imit ;  if  divided  by  10,  diminished  by  a  unit. 


That  is,  the  Log. 

of  5547                 being 

3.744058, 

(( 

(I 

554.7                   is 

2.744058, 

u 

(( 

55.47 

i:  744058, 

(t 

« 

6.547 

0.744058, 

ft 

« 

.5547 

1.744058, 

t( 

« 

.05547 

2.744058, 

« 

(I 

.005547 

3.744058. 

LOGARITHMS. 


Hence,  the  decimal  part  of  the  Logarithm  of  any  set  of  figures 
is  the  same,  wherever  the  decimal  point  may  he. 

The  miims  sign  is  written  over  the  characteristic,  as  only 
the  characteristic  is  negative. 


TABLE  OF  LOGARITHMS. 

4.  In  the  tables  of  Logarithms,  generally,  the  decimal  only 
is  given,  and  the  characteristic  must  be  supplied,  according  to 
the  Rule  in  Art.  2. 

5t  To  find  the  Logarithm  of  any  number  of  three  or  less 
figures. 

Find  the  given  number  in  the  column  marked  N.,  and 
directly  opposite,  in  the  column  marked  0,  is  the  decimal 
part  of  the  Logarithm,  to  which  must  be  prefixed  the 
characteristic,  according  to  the  Rule  in  Art.  2. 

E.  g.  The  Log.  of  832    is    2.920123 

"      108     "    2.033424 

The  first  two  figures  of  the  decimal,  remaining  the  same 
for  several  successive  numbers,  are  not  repeated,  but  are  left 
to  be  supplied.     Thus  the  Log.  of  839  is  2.923762. 

As,  according  to  Art.  3,  multiplying  a  number  by  10  in- 
creases its  Logarithm  by  a  unit,  therefore,  to  find  the  Logarithm 
of  any  number  containing  only  three  significant  figures  with 
one  or  more  ciphers  annexed,  we  use  the  same  rule  as  in 
the  last  case. 

E.  g.  The  Log.  of      8320     is     3.920123 

"     756000      "     5.878522 

The  Logarithms  of  the  integral  numbers  from  1  to  100  inclu- 
sive are  given  with  the  characteristic  on  the  first  page  of  the 
tables. 


4  PLANE  TRIGONOMETRY. 

6*  To  find  the  Logarithm  of  any  number  consisting  of  four 
figures. 

Look  for  the  first  three  figures  in  the  column  marked  N., 
and  for  the  fourth  figure  at  the  top  of  one  of  the  columns. 
Opposite  the  first  three  figures,  and  in  the  column  under  the 
fourth  figure,  will  be  the  last  four  figures  of  the  decimal  part 
of  the  Logarithm,  to  which  the  first  two  figures  in  the  column 
marked  0  are  to  be  prefixed,  and  the  characteristic,  according 
to  the  Rule  in  Art.  2.  As  shown  in  Art.  3,  moving  the  deci- 
mal point  of  a  number  to  the  right  increases,  to  the  left  de- 
creases, the  characteristic  as  many  units  as  the  number  of 
places  the  point  is  moved.  In  some  of  the  columns  marked  1, 
2,  3,  &c.,  dots  will  be  found.  This  shows  that  the  two  figures 
which  arc  to  be  prefixed  from  the  column  marked  0  have 
changed  to  the  next  larger  number,  and  are  to  be  found  in  the 
horizontal  line  directly  below.  The  dots  are  used  to  avoid  any 
mistake,  and  their  place  is  to  be  supplied  with  ciphers. 

E.  g.  The  Log.  of      2951     is     3.469969 

5496     "     3.740047 
"     768700     "     5.885757 

7.  To  find  the  Logarithm  of  any  number  consisting  of  more 
than  four  figures. 

Find  the  Logarithm  of  the  first  four  figures  as  before  ;  mul- 
tiply by  the  remaining  figures  the  number  standing  opposite, 
in  the  column  marked  D.,  reject  from  the  right  as  many  figures 
as  you  multiply  by,  and  add  what  is  left  to  the  Logarithm  pre- 
viously found. 

E.  g.     Required,  the  Log.  of  609946. 
The  Log.  of  609900  is  5.785259 

Under  D.  opposite  is  71,  which  multiplied  by  46  gives        32.66 
Therefore,  the  Log.  of  609946  is  5.785292  * 

*  Whenever  the  fractional  part  omitted  is  larger  than  half  the  unit  in  the 
next  place  to  the  left,  one  is  added  to  that  figure. 


LOGARITHMS.  6 

Required,  the  Log.  of  84997. 
Log.  of  84990  is  4.929368 

Under  D.  opposite  is  51,  which  multiplied  by  7  gives  36.7 

Therefore,  Log.  of  84997  is  4.929404 

The  column  marked  D.  contains  the  average  difference  of  the 
ten  Logarithms  against  which  it  stands.  The  reason  for  reject- 
ing from  the  product  as  many  figures  as  you  multiply  by  is, 
that  these  figures  are  just  so  many  places  farther  to  the  right 
than  the  figures  whose  Logarithm  has  already  bejn  found. 

This  method  of  finding  the  Logarithms  of  large  numbers  sup- 
poses that  the  Logarithms  vary  as  the  numbers,  which  is  not 
strictly  true,  though  sufficiently  so  to  allow  the  use  of  this 
method  in  ordinary  calculations. 

If  the  number  whose  Logarithm  is  sought  contains  decimal 
figures,  the  decimal  part  of  the  Logarithm,  according  to  Art.  3, 
is  the ^ same  as  though  there  were  no  decimal  point;  but  the 
characteristic  varies  according  to  the  Rule  in  Art.  2. 

The  Logarithm  of  a  vulgar  fraction  may  best  be  found  by 
reducing  the  fraction  to  a  decimal,  and  then  proceeding  as 
above. 

8.  To  find  the  Natural  Number  corresponding  to  a  given  Log- 
arithm. 

Neglecting  the  characteristic,  find,  if  possible,  in  the  table 
the  Logarithm  given.  The  three  figures  opposite  in  the  column 
N.,  with  the  number  at  the  head  of  the  column  in  which  the 
Logarithm  is  found,  affixed,  and  the  decimal  point  so  placed  as 
to  make  the  number  of  integral  figures  correspond  to  the  char- 
acteristic of  the  given  Logarithm,  as  taught  in  Art.  2,  will  be 
the  number  required. 

E.  g.  The  Natural  Number  corresponding  to  Log.  5.531862 
is  340300. 

The  Natural  Number  corresponding  to  Log.  1.605951  is 
40.36. 

If  the  decimal  part  of  the  Logarithm  cannot  be   exactly 


6  PLANE  TRIGONOMETKY. 

found,  take  the  Natural  Number  corresponding  to  the  next  less 
Logarithm,  as  before ;  then  find  the  difference  between  this  and 
the  given  Logarithm;  divide  this  difference  by  the  tabular 
difference  in  the  column  opposite,  under  D.,  annexing  to  the 
dividend  one  cipher  to  get  the  first  fi,gure  of  the  quotient,  and 
a£lx  this  quotient  to  the  number  already  found. 

E.  g.  Required,  the  Natural  Number  corresponding  to 
Log.  2.763598 

Next  less  Log.,  2.763578,  and  number  corresponding,  580.2 

Divide  by ; ?orD.%pp"  \  75)20.0(267  267 

Number  required,  580.2267 

The  number  corresponding  to  Log.  4.816601  will,  in  the 
same  way,  be  found  to  be  .000655542. 

Examples. 

1.  Find  the  Log.  of  3764. 

2.  Find  the  Log.  of  2576000. 

3.  Find  the  Log.  of  7.546. 

4.  Find  the  Log.  of  0.0017. 

5.  Find  the  Log.  of  -fj. 

6.  Find  the  Natural  Number  to  Log.  3.807873. 

7.  Find  the  Natural  Number  to  Log.  1.820004. 

8.  Find  the  Natural  Number  to  Log.  2.982197. 

9.  Find  the  Natural  Number  to  Log.  2.910037. 
10.  Find  the  Natural  Number  to  Log.  4.850054. 

9»  The  great  utility  of  Logarithms  in  arithmetical  opera^ 
tions  consists  in  this,  that  addition  takes  the  place  of  mul- 
tiplication, and  subtraction  of  division,  multiplication  of  in- 
volution, and  division  of  evolution.  That  is,  to  multiply 
numbers,  we  add  their  Logarithms ;  to  divide,  we  subtract  the 
Logarithm  of  the  divisor  from  that  of  the  dividend ;  to  raise  a 
number  to  any  power,  we  multiply  its  Logarithm  by  the  expo- 
nent of  that  power ;  and  to  extract  the  root  of  any  number,  we 


LOGARITHMS.  7 

divide  its  Logarithm  by  the  number  expressing  the  root  to  be 
found. 

This  is  the  same  as  multiphcation  and  division  of  different 
powers  of  the  same  letter  by  each  other  in  Algebra,  and  in- 
volving and  evolving  powers  of  a  single  letter  or  quantity ;  the 
number  10  takes  the  place  of  the  given  letter  in  Algebra,  and 
the  Logarithms  are  the  exponents  of  10. 

/    ft  ir 

A-"^       MULTIPLICATION  BY  LOGARITHMS.        C£Xa«^         , 

10.  Rule.  Add  the  Logarithms  of  the  factors,  and  the  sum 
toill  he  the  Logarithm  of  the  product. 

1.  Multiply  347.676  by  475.2.  Ans.  165215.6352. 

2.  Find  the  product  of  568,  7496,  846,  and  1728. 

Ans.  6224314285714. 

(It  must  be  carefully  borne  in  mind  that  the  decimal  part  of 
the  Logarithm  is  always  positive.) 

3.  Multiply  0.00756  by  17.5. 

Log.  of  0.00756  3.878522 

17.5  1.243038 

Product,  0.1323.  Log.  1.121560 

4.  Multiply  0.0004756  by  1355. 

Although  negative  quantities  have  no  Logarithms,  yet,  since 
the  numerical  product  is  the  same  whether  the  factors  are  posi- 
tive or  negative,  we  can  use  Logarithms  in  multiplying  when 
one  or  more  of  the  factors  are  negative,  taking  care  to  prefix  to 
the  product  the  proper  sign  according  to  the  rules  of  Algebni. 
When  a  factor  is  negative,  to  the  Logarithm  which  is  used  n  is 
appended.  'J 

E.  g.   6.    Multiply  —75.46  by  54.5.        '^ 

Log.  of   —75.48  1.877717  n 

54.5  , 1.736397 

Product,  —4112.57.  Log.  3.614114  n 


8 


PLANE   TKIGONOMETRy. 


(Log.  of — 76.46,  though  incorrect,  is  used  for  the  sake  of 
brevity.) 

6.    Find  the  product  of —0.017^25,  and  —165.4. 


Find  the  product  of —14,  —7.643,  and  —0.004. 

Ans.  —.428008 


DIVISION  BY  LOGARITHMS. 


ip^ 


\> 


11 »  Rule.  From  the  Logarithm  of  the  dividend  subtract 
the  Logarithm  of  the  divisor,  and  the  remaiiider  will  he  the 
Logarithm  of  the  quotient. 

E.  g.  1.    Divide  78.46  by  0.00147. 


Log.  of  78.46 

0.00147 

1.894648 
3.167317 

Quotient,  53374.1. 
2,   Divide  0.0014  by  756. 

Log.  4.727331 

,5>4^''               Log.  of       0.0014 
Va^^                        "       756 
Quotient,  0.000001852. 

3.146128 

2.878522 
Log.  6.267606 

Negative  numbers  can  be  divided  in  the  same  manner  as 
positive,  taking  care  to  prefix  to  the  quotient  the  proper  sign, 
according  to  the  rules  of  Algebra. 

3.  Divide  .7478  by  0.00456. 

4.  Divide  5000  by  0.00149. 
6.    Divide  0.00997  by  64.16. 

6.  Divide  —14.55  by  543. 

7.  Divide  —465  by  —19.45. 


Ans.  163.99+. 

Ans.  0.00015539+. 

Ans.  — 0.0267955-I-. 

Ans.  23.9074-I-. 


INVOLUTION  BY  LOGARITHMS. 

12.    Rule.     Multiply   the   Logarithm  of  the  number  by  the 
exponent  of  the  power  required. 

1.    Find  the  15th  power  of  1.17. 
Log.  of  L17 


Ans.  10.638. 


0.068186 

15 

Log.  1.022790 


LOGARITHMS. 

2.    Find  the  5th  power  of  0.00941. 

Log.  of  0.00941  3.973590 

5 


Ans.  0.000000000073782.  Log.  11.867950 

3.  Fipd  the  4th  power  of  0.0176.  Ads.  0.000000095951+. 

4.  F'md  the  9th  power  of  1.179.  Ans.  4.401765+. 
Negative  numbers  are  involved  in  the  same  manner,  taking 

care  to  prefix  to  the  power  the  proper  sign,  according  to  the 
rules  of  Algebra. 

5.  Find  the  3d  power  of —0.017.        Ans.  —0.000004913. 

6.  Find  the  6th  power  of —14.  Ans.  7529536. 

EVOLUTION  BY  LOGARITHMS. 

13«  Rule.  Divide  the  Logarithm  of  the  number  by  the  expo- 
nent of  the  root  required. 

Negative  numbers  are  evolved  in  the  same  manner,  taking 
care  to  prefix  to  the  root  the  proper  sign,  according  to  the  rules 
of  Algebra.  For  the  sake  of  convenience,  where  the  character- 
istic of  a  Logarithm  is  negative,  and  not  divisible  by  the  index 
of  the  root,  we  can  increase  the  negative  characteristic  so  as  to 
make  it  divisible,  providing  we  prefix  an  equal  positive  number 
to  the  decimal  part  of  the  Logarithm. 

E.  g.  1.   Find  the  5th  root  of  0.0173. 

Log.  of  0.01 73  is  2.238046,  which  is  equal  to  5  +  3.238046,  and 
dividing  this  by  5  gives  1.647609,  which  is  the  Log.  of  0.4442. 

2.  Find  the  3d  root  of  80.07.  Ans.  4.31013+ 

3.  Find  the  8th  root  of  0.0764.  Ans.  .72508+ 

4.  Find  the  7th  root  of —17.  Ans. —1.49891+ 

5.  Find  the  5th  root  of  —0.00496.  Ans.  —0.34601+ 

lit  Instead  of  subtracting  one  Logarithm  from  another, 
it  is  sometimes  more  convenient  to  add  what  it  lacks  of 
10,  —  which  difference  is  called  the  complement,  —  and  from 


.11 


10  PLANE  TRIGONOMETRY. 

the  sum  reject  10.  The  result  is  evidently  the  same.  For 
X  —  y  =  a?-|-(10  —  y)  —  10.  The  complement  is  easiest  found 
by  beginning  at  the  left  of  the  Logarithm  of  the  number,  and 
subtracting  each  figure  from  9,  except  the  last  significant  fig- 
ure, which  must  be  subtracted  from  10. 
.    In  proportion,  therefore,  we  have  the  following  rule  : 

Add  the  complement  of  the  Logarithm  of  the  first  term  to  the 
Logarithms  of  the  second  and  third  terms,  and  from  the  sum 
reject  10. 

E.  g.  1.    Find  a  fourth  proportional  to  14,  175,  and  7486. 

Complement  of  Log.  of      14,  8.853872 

175,  2.243038 

7486,  3.874250 

Ans.  93575.  Log.  4.971160 

2.  Given  the  first  three  terms  of  a  proportion,  416,  584, 
and  256,  to  find  the  fourth.  Ans.  359.38+. 

3.  Find  the  value  of  179  X  4968  -r-  489. 

Ans.   1818.552+. 

4.  Find  the  value  of  ^1-^^^}1.  Ans.  4.7776+ 

d04  X  513 

6.  Find  the  value  of V(0.1739) _^ 

331.9  (v/2.04  +  V^l.203/ 

Ans.  0.000197055. 

23.3X6.764   X^ 
6.    Find  the  value  of  "^''^-^  - 


X9.97 


Ans.  838.965+ 

7.  In  a  system  whose  base  is  4,  what  is  the  Logarithm  of 
41  of  161  of  641  of  21  of  81  of  1 1  ofjl  of  J1  of  J1  ofOI 

8.  Solve  the  equation  125*  =  25. 

X  X  Log.  125  =  Log.  25 

_  Locr.  25   __  1.39794  _  2     . 
^  —  Log.  125  ~  2:09691  ~  3'  ^^^* 

9.  Solve  the  equation  2048*  =  16. 


TRIGONOMETRIC  FUNCTIONS. 


11 


CHAPTER  II. 


TRIGONOMETRIC   FUNCTIONS. 


DEFINITIONS. 

15.  Trigonometry  is  that  branch  of  mathematics  which  treats 
of  methods  of  computing  angles  and  triangles. 

16.  Plane  Trigonometry  treats  of  methods  of  computing 
plane  angles  and  triangles. 

17.  The  circumference  of  every  circle  is  divided  into  360 
equal  parts,  called  degrees  ( °  )/ea:clr  degree  into  60  equal  parts, 
called  minutes  ( ' ),  and  each  minute  into  60  equal  parts,  called 
seconds  ( '' ). 

18.  As  angles  at  the  centre  vary  as  their  arcs,  or  arcs 
as  their  corresponding  angles,  the  measure  of  an  angle  is  the 
arc  included  between  its  sides  and  described  from  its  vertex  as 
a  centre  (Geom.,  III.  14). 

19.  As  the  sum  of  all  the  angles  about  the  point  C  is  equal 
to  four  right  angles,  one  right  angle,  A  C  B,  would  be  measured 
by  one  quarter  of  the  circumference,  or  90°  (Geom.,  III.  15). 

20.  The  Complement  of  an  arc  or 
angle  is  90°  minus  this  arc  or  angle. 
Thus,  the  arc  AD  is  the  complement 
of  D  B,  and  the  angle  A  C  D  of  D  C  B. 
When  an  arc  or  angle  is  greater  than 
90°,  its  complement  is  negative. 

21.  The  Supplement  of  an  arc  or 
angle  is  180°  minus  this  arc  or  angle. 
Thus,  the  arc  EAD  is  the  supplement 

of  D  B,  and  the  angle  E  C  D  of  D  C  B.     When  an  arc  or  angle 
is  greater  than  180,°  its  supplement  is  negative. 


12 


PLANE   TKIGONOMETRY. 


A 


22»  *  The  Sine  of  an  arc  or  angle 
is  the  line  drawn  from  one  end  of  the 
arc,  perpendicular  to  the  diameter  pass- 
ing through  the  other  end;  or  it  is 
half  the  chord  of  double  the  arc.  Thus, 
D  F  is  the  Sine  of  the  arc  D  B,  or  of 
the  angle  D  C  B. 

23.   The  Versed  Sine  of  an  arc  or 
angle  is  that  part  of  the  diameter  which 
is  between  the  foot  of  the  sine  and  the  arc.     Thus,  B  F  is  the 
Versed  Sine  of  the  arc  B D,  or  of  the  angle  BCD. 

24*  The  Cosine  of  an  arc  or  angle  is  the  sine  of  the  com- 
plement of  the  arc  or  angle,  or  the  radius  minus  the  versed 
sine  of  the  arc  or  angle.  Thus,  H  D  =  C  F  is  the  Cosine  of 
the  arc  B D,  or  of  the  angle  BCD. 

(Co  in  Cosine,  &c.,  stands  for  complement.) 

25.  The  Tangent  of  an  arc  or  angle  (in  Trigonometry)  is 
tbe  line  touching  one  extremity  of  the  arc,  and  terminated  by 
a  line  drawn  from  the  centre  through  the  other  extremity. 
Thus,  B I  is  the  Tangent  of  the  arc  B  D,  or  of  the  angle 
BCD. 

26.  The  Cotangent  of  an  arc  or  angle  is  the  tangent  of  the 
complement  of  the  arc  or  angle.  Thus,  A  K  is  the  Cotangent 
of  B D,  or  of  the  angle  BCD. 

27.  The  Secant  of  an  arc  or  angle  (in  Trigonometry)  is  the 
line  drawn  from  the  centre  through  one  end  of  the  arc,  and 
terminated  by  the  tangent  to  the  other  end.  Thus,  C I  is  the 
Secant  of  B D,  or  of  the  angle  BCD. 

28.  The  Cosecant  of  an  arc  or  angle  is  the  secant  of  the 
complement  of  the  arc  or  angle.  Thus,  C  K  is  the  Cosecant  of 
B D,  or  of  the  angle  BCD. 


*  Those  who  prefer  the  Analytical  Method  will  turn  from  this  point  to 
Chapter  IV. 


/J 


TRIGONOMETRIC  FUNCTIONS.  13 

29.  The  Sine,  Tangent,  and  Secant  of  the  supplement  of 
an  arc  are  (irrespective  of  the  signs)  the  same  as  for  the  arc 
itself. 

The  Sine  and  Cosine  of  an  arc  form  the  two  sides  of  a  right- 
angled  triangle  whose  hypothenuse  is  the  Radius  of  the  arc. 

The  Radius  and  Tangent  of  an  arc  form  the  two  sides  of 
a  right-angled  triangle  whose  hypothenuse  is  the  Secant  of  the 
arc. 

30i  Suppose  the  Radius  C  D  to  move  in  the  plane  of  the 
circle  about  the  centre  C  :  let  it  move  so  that  the  arc  'B  D  and 
the  angle  BCD  become  0 ;  then  the  Sine,  Tangent,  and  Versed 
Sine  of  the  arc  or  angle  become  0;  the  Secant  and  Cosine 
equal  to  Radius ;  the  Cosecant  and  Cotangent  infinite. 

If  C  D  moves  toward  A  until  the  arc  B  D  or  the  angle  BCD 
becomes  30°,  then,  if  Radius  is  unity, 

D  F,  or  Sine  30°= J 
For  the  Sine  D  F  is  half  the  chord  of  double  the  arc,  that  is,  is 
half  of  D  G,  which,  as  it  subtends  sixty  degrees,  or  one  sixth  of 
the  circumference,  is  equal  to  Radius  (Geom.,  III.  33). 

If  C  D  moves  until  the  arc  B  D  or  the  angle  BCD  becomes 
45°,  then  the  triangle  CDF  becomes  isosceles ;  and  if  Radius 
is  unity,  we  have 

FD^  +  FC^  =  2  DF'' ==  C  D^ 
Hence  D  F  =  C  D  y^J 

or  Sine  45°  =  v'^  =  ^V'2 

The  Tangent,  in  this  case,  equals  the  Radius. 

If  C  D  moves  until  B  D,  or  B  C  D,  becomes  60°,  then  since 


DF  =  VCD2  — CF2 
and   C  F  =  Sine  of  A  D  =  Sine  30°  =  ^ 

D  F,  or  Sine  60°  =  V^f^  =  y'f  =  ^  y/S 

If  C D  moves  until  the  arc  B D,  or  the  angle  BCD,  becomes 
90°,  then  the  Sine,  Versed  Sine,  and  Cosecant  become  equal  to 
Radius ;  the  Cosine  and  Cotangent  0 ;  the  Secant  and  Tangent 
infinite. 


14 


PLANE  TRIGONOMETRY. 


If  we  suppose  C  D  to  move  until  the  point  D  passes  entirely 
round  the  circumference,  it  will  be  easy  to  trace  the  changes  in 
the  length  of  the  Sine,  Cosine,  &c. 

31 1    The  centre  C  is  the  absolute  zero  point. 

If  we  consider  a  line  extending  in 
one  direction  plus,  a  line  extending  in 
the  opposite  direction  should  be  con- 
sidered minus.  It  has  been  agreed 
to  consider  the  trigonometric  lines 
which  extend  from  EB  upward,  or 
from  A  L  to  the  right,  plus ;  therefore 
all  those  extending  from  B  E  down- 
ward, or  from  A  L  to  the  left,  must  be 
considered  minus.  It  will  on  inspection  be  found  that  these 
lines  change  their  direction  at  the  point  where  they  become 
0,  or  infinite.  Therefore,  the  algebraic  signs  of  the  Sines,  Co- 
sines, &c.,  change  from  plus  to  minus,  or  minus  to  plus,  as  each 
passes  the  point  where  it  becomes  0,  or  infinite.  These  changes 
in  the  signs  will  be  found  to  be  as  follows  : 


1st  quadr. 
Sine  and  Cosecant  -f- 

Cosine  and  Secant  -f- 

Tangent  and  Cotangent      -\- 


2d  quadr. 

+ 


3d  quadr.     4th  quadr. 


+ 


+ 


32.  By  -various  methods,  the  Sines,  Cosines,  Tangents,  and 
Cotangents  have  been  calculated  for  every  minute  of  the 
Quadrant,  with  Radius  as  unity ;  and  the  logarithms  of  these 
numbers  have  been  taken  from  the  table  of  logarithms,  and, 
with  10  added  to  the  characteristic,  to  avoid  negative  charac- 
teristics (that  is,  the  radius  assumed  is  10000000000),  have 
been  arranged  in  the  table  entitled  Logarithmic  Sines  and 
Tangents. 

33*  To  find  the  Logarithmic  Sine,  Cosine,  &c.,  of  any  arc  or 
angle. 

In  the  tables  the  degrees  up  to  45°  are  at  the  top  of  the  page, 


TKIGONOMETRIC   FUNCTIONS.  15 

and  the  minutes  on  the  left ;  above  45°  (since  the  Sine,  or  Tan- 
gent, of  any  arc  is  the  Cosine,  or  Cotangent,  of  its  complement), 
the  degrees  are  at  the  bottom  of  the  page,  and  the  minutes  on 
the  right.  In  the  first  and  second  columns,  marked  D.,  is  the 
rate  of  variation  per  second  for  the  columns  at  their  left,  and  in 
the  third,  marked  D.,  the  rate  of  variation  for  the  columns  on 
both  sides  of  it.  In  the  columns  marked  D.  the  last  two  figures 
are  to  he  considered  as  decimals. 

It  must  be  remembered  that,  as  the  arc  or  angle  increases, 
the  Sines  and  Tangents  increase,  while  tlie  Cosines  and  Co- 
tangents decrease. 

E.  g.    The  Log.  Sine  of  37°  10'  is  9.781134 

74°  50'  '*  9.984603 

1.  Requires,  the  Log.  Sine  of  41°  14'  25''. 

Log.  Sine  of  41°  14'  is  9.818969 

Number  to  be  added  for  2^"  is  25  X  2.4  =  60 

Ans.  9.819029 

2.  Required,  the  Log.  Cosine  of  65°  24'  b". 

Log.  Cosine  of  65°  24'  is  9.619386 

Number  to  be  subtracted  for  5"  is  5  X  4:. 6  =  23 

;    IL    n  r    '  Ans.  9.619363 

31*  To  find  the  degrees,  minutes,  and  seconds  corresponding 
to  any  Logarithmic  Sine,  Cosine,  &c. 

.  Find  in  the  column  with  the  given  title  (that  is.  Sine,  Cos., 
Tan.,  or  Cot.)  the  given  logarithm ;  if  the  title  is  at  the  top, 
take  the  degrees  at  the  top  and  the  minutes  on  the  left ;  but  if 
the  title  is  at  the  bottom,  take  the  degrees  at  the  bottom  and 
the  minutes  on  the  right.  If  the  given  logarithm  is  not  found 
exactly,  take  the  degrees  and  minutes  corresponding  to  the 
next  less  logarithm  for  Sines  and  Tangents,  next  greater  for 
Cosines  and  Cotangents ;  divide  the  difference  of  these  two 
logarithms  by  the  corresponding  tabular  difference  D.,  and  the 
quotient  will  be  the  additional  number  of  seconds. 


16  PLANE  TRIGONOMETRY. 

E.  g.   1.    Eequited,  the  degrees,  minutes,  and  seconds  corre- 
sponding to  the  Log.  Sine  9.874321. 
9.874321 
Log.  Sine  48°  28'',       9.874232 

1.87)89.00 

48         Ans.  48°  28'  48". 

2.  Eequired,  the  degrees,  &c.  corresponding  to  Log.  Cotan- 
gent 9.911302. 

Log.  Cotangent  50°  48',  9.911467 
9.911302 

4.3)165.0 

38        Ans.  50**  48'  38". 

3.  Find  the  Log.  Sine  of  13°  10'  31". 

4.  Find  the  Log.  Sine  of  76°  10'  49". 

6.  Find  the  Log.  Cosine  of  87°  51'  42". 
6r  Find  the  Log.  Cosine  of  175°  43'  44". 

7.  Find  the  Log.  Cotangent  of  17°  16'  14". 

8.  Find  the  Log.  Cotangent  of  49°  15'  27". 

9.  Find  the  Log.  Tangent  of  43°  5^  44". 

10.  Find  the  Log.  Tangent  of  113°  21'  5". 

11.  Given  Log.  Sine  8.898611,  to  find  the  degrees,  &c.  cor- 
responding. 

12.  Given  the  Log.  Tangent  9.47864,  to  find  the  degrees,  &c. 
corresponding. 

13.  Given  the  Log.  Sine  9.90543,  to  find  the  degrees,  &c.  cor- 
responding. 

14.  Given  the  Log.  Cosine  9.996087,  to  find  the  degrees,  &c. 
corresponding. 

15.  Given  the  Log.  Cosine  9.846321,  to  find  the  degrees,  &c. 
corresponding. 

16.  Given  the  Log.  Cotangent  10.5673  to  find  the  degrees, 
&c.  corresponding. 

17.  Given  the  Log.  Cotangent  9,  to  find  the  degrees,  &c.  cor- 
responding. 


SOLUTION   OF  PLANE  TRLAJ^GLES.  17 

CHAPTER  III. 

SOLUTION   OF   PLANE   TRIANGLES. 

35*  In  every  plane  triangle  there  are  six  parts,  three  sides 
and  three  angles.  Of  these,  any  three  being  given,  provided 
one  is  a  side,  the  others  can  be  found. 

RIGHT-ANGLED  TRIANGLES. 

36*  In  a  right-angled  triangle,  one  of  the  six  parts,  viz.  the 
right  angle,  is  always  given;  and  if  one  of  the  acute  angles 
is  given,  the  other  is  known;  therefore,  in  a  right-angled 
triangle,  the  number  of  parts  to  be  considered  is  four,  any  two 
of  which  being  given,  the  others  can  be  found.  We  may  have 
four  cases,  according  as  there  are  given, 

1.  The  hypothenuse  and  an  acute  angle  ; 

2.  A  side  about  the  right  angle  and  an  acute  angle ; 

3.  A  side  about  the  right  angle  and  the  hypothenuse  ; 

4.  The  sides  about  the  right  angle. 

All  these  cases  can  be  solved  by  the  following  Theorem  : 

THEOREM  I. 

37  •    In  any  right-angled  plane  triangle, 

Ist.  Radius  is  to  the  hypothenuse  as  the  sine  of  either  acute 
angle  is  to  the  opposite  side; 

2d.  Radius  is  to  either  side  as  the  tangent  of  the  adjacent 
acute  angle  is  to  the  opposite  side. 

Let  ABC  be  a  triangle,  right- 
angled  at  C.  Let  h  represent  the 
hypothenuse,  and  a  and  h  the  sides 
opposite  the  angles  A  and  B  respec- 
tively. With  either  angle,  as  A,  as 
a  centre,   and   any  radius,    (which 


18 


PLANE  TKIGONOMETEY. 


radius  will  represent  the  radius  of 
the  tables,)  describe  the  arc  DE; 
from  D  draw  D  F  perpendicular  to 
A  e,  and  draw  E  G  parallel  to  D  F. 
Then  D  F  will  be  the  tabular  sine,  A 
and  G  E  the  tabular  tangent  of  the 
angle  A. 

From   the  similar   triangles   ADF,    AGE,  and  ABC,  we 
have, 


1st. 

AD  :  AB  =  DF  :  BC 

b 

^ 

that  is, 

R  :  h=z  sin.  A  :  a 

a. 

2d. 

AE  :  ACr=GE  :  BC 

-x 

that  is, 

R  :  6  =  tan.  A  :  a 

• 

38. 

Corollary  1. 

As 

sin.  A  =  cos.  B 
R  :  A  =  COS.  B  :  a 

39.    Corollary  2.    If  radius  is  unity  these  proportions  will 
give, 

a -==.11  sin.  A  a  =  b  tan.  A         a=.h  cos.  B 


Case  I. 

40.    Given   the   hypothenuse    and  an  acute 
angle.  / 

From  Theorem  I. 

R  :  A  =  sin.  A  :  a ' 
R  :  A  =  sin.  B,  or  cos.  A  :  h 
Ex.  1.    Given  h  255,  A  57°  14',  to  find  a    a 
and  h. 


By  Logarithms, 

Radius 

10. 

:A255 

2.406540 

=  sin.  A  57°  W 

9.924735 

:  a  214.42 

2.331275 

SOLUTION  OF  PLANE  TRIANGLES.  19 

Radius  10. 

:  h  255  2.406540 

Oj^  COS.  A  57°  14'  9.733373 

"0    :  6  138.01  2.139913 

Ud  n    Ex52.    Given  h  1676,  A  67°  13',  to  find  a  and  h. 

{  a  1545.23. 


Ans. 

h    649.03. 


Ex.  3.    Given  h  78.4,  B  15°  51',  to  find  a  and  h 

Ans. 


f  a  75.42. 

/7-X./"""'    16  21.41. 

Case  II.  '  "-  *7 


41*    6^m?i  a  sic?e  about  the  right  angle  and  an  a^ute  angle. 
From  Theorem  I. 

sin.  A  :  a  =  R  :  A 

R  :  A  =  sin.  B  :  6  U    ^  ^ 

Ex.  1.    Given  a  195,  B  64°  43',  to  find  h  and  h, 

sin.  A  =  cos.  64°  43'        comp.     0.369476 

2.290035 

2.659511 

10. 

2.659511 
9.956268 
2.615779 

Ex.  2.    Given  h  1075,  B  75°  49',  to  find  a  and  h. 

^^g     ia    271.68. 
\  h  1108.79. 

Ex.  3.    Given  6  17.45,  A  47°  31',  to  find  a  and  h. 

^g     I  a  19.05. 
^'    I  A  25.84. 


'.a  195 

R 

:  h  456.57 

R 

:A 

=  sin.  B  64° 

43' 

:  6  412.84 

20 


PLANE  TRIGONOMETRY. 


Case  III. 

42.    Given  a  side  about  the  right  angle  and 
the  hypothenuse. 
From  Theorem  I. 

A  :  R  =  a  :  sin.  A 
R  :  A  =  sin.  B  :  h 

Ex.  1.    Given  h  24.5,  a  17.4,  to  find  the  other  parts. 


h  24.5 
:R 
=  a  17.4 

comp.     8.610834 
10. 
1.240549 

:  sin  A  45°  15' 

5'' 

9.851383 

B  =  90°  — 

45°  15' 

5"  =  44°  44'  55" 

R 
:  h  24.5 

=  sin.  B  44°  44' 

55'' 

10. 
1.389166 
9.847571 

.    :  6  17.248 

1.236737 

Otherwise  h  can  be  found  from  the  formula  b^  =  h^  —  a*. 

.-.  b  = 
Ex.2.   Given  A  172.8, 

b  14.17, 

a)  (A  -  a) 

to  find  the  other  parts. 

/-A  85°  17' 47". 
Ans.   ^  B    4°  42'  13". 
1  a  172.218. 

Case  IV. 
43  •    Given  the  sides  about  the  right  angle. 
From  Theorem  I. 

b  :  R  =  a  :  tan.  A 
sin.  A  :  a  =  R  :  A 

Ex.  1.    Given  a  195,  6  147,  to  find  the  other  parts. 
b  147  comp.    7.832683 

:R  10. 

=  al95  2.290035 

•.  tan.  A  52°  69'  22"  10.122718 


SOLUTION   OF  PLANE  TRIANGLES.  21 

B  =  90°  —  52°  59'  22''  =  37°  0'  38^ 

sin.  A  52°  59'  22''  comp.    0.097712 

:  a  195  2.290035 

=  R         .  lO 

:  h  244.2  2.387747 

Otherwise  h  can  be  found  from  the  formula  h^  =  a^  -\-  h^ ; 
then  the  angles  by  using  the  first  proportion  of  Theorem  I. 


Ex.  2.    Given  a  189,  h  14,  to  find  the  other  parts. 

A  85°  45'  49^. 
Ans.    i  B    4°  14' 11". 
189.518. 


44*   In  the  following  Examples  two  parts  of  a  right-angled 
triangle  are  given,  and  the  others  required. 

a  888.896. 
1.   Given  6  217,   A  915.  Ans.    ^  A  76°  16' 52". 


3. 


'If 

Given  a  17.94,   A  15°  39'.  Ans.    | 


B  13°  43'   8". 

A    5°    9' 34". 
2.   Given  a  174,   h  1927.  Ans.    \  B  84°  50'  26". 

1934.89. 


h  64.038. 
h  66.503. 


4.  Given  ^  47.9,   A  59°  17'.  Ans.    (  «  ^1'1'^^8. 

I  h  24.467. 

A  23°  36'  42'; 

5.  Given  a  298,   h  744.  Ans.    {  B  66°  23'  18". 


If 


681.712. 


A  35°  57'  32". 
6.    Given  a  9.75,    6  13.44.  Ans.   {B  W"    2' 28". 

h  16.60. 

( 
03042. 


7.   Given  6  0.02518,   A  34°  7' 10".         Ans.    /^^-^^^^^ 

I  h  0.030 


^ 


22Ns^/^  PLANE  TRIGONOMETRY. 


OBLIQUE-ANGLED  TRIANGLES. 

45*    In  solving  oblique-angled  triangles,  there  are  four  cases 
There  may  be  given, 

1.  Two  angles  and  a  side  ; 

2.  Two  sides  and  an  angle  opposite  one  of  them  ; 

3.  Two  sides  and  the  included  angle  • 

4.  The  three  sides. 

For  solving  these  we  demonstrate  the  three  following  Theorems. 

THEOEEM  II. 

46.    In  dny  filane  triangle^  the  sides  have  the  same  ratio  as  the 
sines  of  the  opposite  angles. 

Let  a,  5,  c  represent  the  sides  oppo- 
site the  angles  A,  B,  C,  respectively. 
Then 

a  :h  '.  €•=.  sin.  A  :  sin.  B  :  sin.  C 

From  B  draw  B  D  perpendicular  to  6. 
Then  A  B  D  and  B  D  C  being  right-angled  triangles,  from  Theo- 
rem I.  we  have 


E  :  a  =  sin.  C  :  B  D 

.-.Ex  BD  =  asin.  C 

R  :c  =  sin.  A  :BD 

.'.Ex  BD  =  csin.  A 

Therefore  a  sin.  C  =  c  sin.  A 

or  a  \  c-=z  sin.  A  :  sin.  C 

In  like  manner  it  can  be  shown  that 

a  :  b  =  sin.  A  :  sin.  B 

h  :  c  =i  sin.  B  :  sin.  C 
Therefore  a  :  6  :  c  =  sin.  A  :  sin.^  B  :  sin.  C 

47.    Scholium.     If  one  of  the  angles,  as  C,  should  become  ;i 


SOLUTION  OF   PLANE  TRIANGLES. 


23 


right  angle,  then  c  will  become  the  hypothenuse,  and  sin.  C 
radius,  and  the  proportion  will  become 

a  :  hz=i  sin.  A  :  R 
or  R  :  A  r=  sin.  A  :  a 

which  is  the  same  as  the  first  proportion  in  Theorem  I. 


THEOREM  III. 

48«  In  any  plane  triangle,  the  sum  of  any  two  sides  is  to  their 
difference,  as  the  tangent  of  half  the  sum  of  the  opposite  angles  is 
to  the  tangent  of  half  their  difference. 

Let  ABC  be  a  plane  triangle ;  then 
BC  +  BA  :  BC  —  B A  =  tan.  ^  {k.  +  Q)  :  tan.  J  (A  —  C) 

Produce  A  B  to  D,  making  B  D  equal 
to  B  C,  and  join  D  C.  Take  B  F  equal 
to  B  A,  draw  A  F  and  produce  it  to  E. 
AD  =  Bc4-BA 
FC  =  BC  — BA 
The  sum  of  the  two  angles  BAF  and 
B  F  A  is  equal  to  the  sum  of  B  A  C  and 
B  C  A,  as  each  sum  is  the  supplement  of 
ABC;  therefore,  as  A  B  is  equal  to  B  F, 

BAF  =  HA  +  C) 
If  from  the  greater  of   two  quantities  we  subtract  half  their 
sum,  the  remainder  will  be  half  their  difference ;  therefore, 

EAC  =  J(A  — C) 
As  B  D  is  equal  to  B  C,  the  angles  B  D  C  and  BCD  are  equal ; 
and  D  A  F  is  equal  to  B  F  A,  and  B  F  A  to  C  F  E  ;  therefore  the 
triangles  A  D  E  and  F  E  C  are  mutually  equiangular ;  hence  the 
two  angles  at  E  are  equal,  and  A  E  is  perpendicular  to  D  C ; 
and  if  with  A  E  as  radius  and  A  as  a  centre,  an  arc  is  de- 
scribed, D  E  becomes  the  tangent  of  D  A  E,  and  EC  of  E  A  C. 
By  similar  triangles  we  have  (Geom.  II.  19) 

AD:FC  =  DE:EC  or 

BC  +  BA  :BC  —  BA  =  tan.  J  (A  +  C)  :  tan.  J  (A  —  C) 


24  PLANE  TRIGONOMETRY. 

THEOREM    IV. 

49i  If  from  any  angle  of  a  plane  triangle  a  perpendicular  he 
drawn  to  the  opposite  side  or  base,  then  the  sum  of  the  segments  of 
the  base  will  be  to  the  sum  of  the  other  two  sides  as  the  difference 
of  these  sides  is  to  the  difference  of  the  segments  of  the  base. 

From   B,    in  the  triangle  ABC,  .^^ 

draw    BD    perpendicular    to   AC.  ^^     \  \ 

Then  ^.^  \      \ 

AD  +  DC:AB  +  BC  =  AB  — BC:AD  — DC 
For         BC2  —  D  C2  =  B  D2  =  AB2  —  AD2 
or  AD2  —  D  C2  __  ^  £2  _  5(^2 

As  the  product  of  the  sum  and  difference  of  two  quantities 
is  equal  to  the  difference  of  their  squares,  we  have 

(A  D  +  D  C)  (A  D  --  D  C)  =  (A  B  +  B  C)  (A  B  —  B  C) 
or     AD  +  DC:AB  +  BC  =  AB— BC:AD  —  DC 

50*  Scholium.  When  the  perpendicular  falls  within  the 
triangle,  the  sum  of  the  segments  of  the  base  is  equal  to  the 
whole  base  ;  when  without,  the  difference.  ' 

Case  I. 

51  •    Given  two  angles  and  a  side. 
From  Theorem  II. 

sin.  A  :  sin.  B  :  sin.  Q  z=  a  :h  :  c 
Ex.  1.   Given  A  48°,  C  55°  17',  a  417,  to 
find  the  other  parts.  ^  & 

B  =  180°  —  (55°  17'  +  48°)  =  76°  43' 

sin.  A  48°  comp.  0.128927 

:  sin.  B  76°  43'  9.988223 

=  a417  2.620136 

:  b  546.12  2.737286 


SOLUTION   OF  PLANE  TRIANGLES.  26 

sin.  A  48°  comp.  0.128927 

:  sin.  C  55**  17'  9.914860 

=  a417  2.620136 

:  c  461^24  2.663923 

Ex.  2.    Given  A  95°  4VB  25°  14',  c  49.17,  to  find  the  other 
parts. 

J  C  59°  42'. 
Ans. 


\  a  56.727. 
i  h  24.278. 


Case  II. 
52*    Given  two  sides  and  an  angle  opposite  to  one  of  them. 
From  Theorem  II. 

a  -.h  :  c  ■=■  sin.  A  :  sin.  B  :  sin.  C 

Ex.  1.    Given  a  55,  c  49.87,  A  25°  44',  to  find  the  other  parts. 

a  55  comp.  8.259637 

:  c  49.87  1.697839 

r=  sin.  A  25°  44'  9.637673 

:sin.  C23°  11' 2"  9.595149 

B  =  180°  —  (23°  11'  2^  +  25°  44')  =  131°  4'  58" 

sin.  C  23°  11'  r  comp.  0.404851 

:  sin.  B  131°  4'  58^  9.877234 

=  c  49.87  1.697839 

:  h  95.483  1.979924 

53*  If  B  C,  the  side  opposite  the 
given  angle,  is  less  than  the  other 
given  side  AB,  and  the  given  angle  a- 
is  acute,  there  are  two  triangles  which 
satisfy  the  conditions,  viz.  ABC  and  ABD,  in  which  the 
angles  B  C  A  and  B  D  A  are  supplements  of  each  other.  The 
Log.  sine  obtained  in  working  such  an  example  represents 
2 


D^--- 


26  PLANE   TRIGONOMETRY. 

either  the  angle  BC  A,  or  its  supplement  BD  A  (Art.  29).  If  the 
given  angle  A  is  obtuse,  or  ths  side  opposite  the  given  angle 
is  greater  than  the  other  given  side,  there  is  but  one  solution 
(Geom.,  VI.  11).  Whenever  the  solution  is  impossible  (Geom., 
VI.  12),  the  Log.  sine  obtained  in  working  the  example  will  be 
greater  than  radius,  which  is  absurd! 

Ex.  2.  Given  a  95.5,  c  173.2,  A  27°  4',  to  find  the  other 
parts. 

/C  55°  36' 47"',  rC  124°  23'  13"^ 

Ans.    }  B  97°  19'  13'^,     or  -|  B    28°  32'  47^ 
(6  208.17,  ib  100.29. 

Case  III. 
54*    Given  two  sides  and  the  included  angle. 
From  Theorem  III. 

a-^c'.a  —  c^  tan.  J  (A  +  C)  :  tan.  |  (A  —  C) 
From  Theorem  II. 

sin.  A  :  sin.  B  .  sin.  Q  ■=  a  \h  :  c 

Ex.  1.    Given  a  976,  c  89,  B  51°  17',  to  find  the  other  parts. 

J  (A  +  C)  =  J  (180°  —  51°  IT)  r=  64°  21'  30'' 

a-\-c=  1065  comp.     6.972650 

:  a  —  ^  =  887  2.947924 

=:tan.  J  (A  +  C)  =  tan.  64°  21'30"  10.318746 

:  tan.  J  (A  —  C)  =  tan.  60°  2'  36"  10.239320 

Half  the  sum  plus  half  the  difierence  gives  the  greater  angle 
A  124°  24'  6";  half  the  sum  minus  half  the  difference,  the 
less  C  4°  18'  54". 

sin.  C  4°  18'  54"  comp.  1.123553 

:  sin.  B  61°  17'  9.892233 

=  c89  1.949390 

:  6  922.94  2.965176 


SOLUTION   OF   PLANE  TRIANGLES.  27 

Ex.  2.    Given  a  91,   b   104,   C   14°  30',  to   find   the    other 
parts. 

rA    55°    5' 37''. 
Ana.    ^B  110°  24' 23^ 
ic  27.783. 

Case  IV. 
55 1    Given  the  three  sides.  B 

From  B  let  fall  a  perpendicular  upon  b.  /i  ^\.  a 

From  Theorem  IV.  /   j  ^^^^^ 

6  :a  +  c  =  a  —  c  :DC  — DA         ^       D       b  ^ 

The  angles  A  and  C  can  then  be  found  as  in  Art.  42. 

Ex.  1.    Given  a  125,  6  135,  c  75,  to  find  the  angles. 
b  135  comp.  7.869666 


:a  +  c 
:DC- 

200 
50 
-  D  A  74.0741 

2.301030 
1.698970 
1.869666 

c 

D  C,  therefore,  is  104.53, 
can  now  be  found. 

and 

DA 

a  A 

30.46 

;  the  angles 
/-A  66°     2' 

A  and 
7"+ 

\th    '.       -^     -  f  i^~-<5    '  )^-fH>  •         ^C  33°  14'  56"+ 

56.  The  sum  of  any  two  sides  must  be  greater  than  the 
remaining  side,  otherwise  the  triangle  is  impossible. 

If  the  perpendicular  is  drawn  to  the  longest  side,  it  will  fall 
within  the  triangle. 

The  shorter  segment  of  the  base  is  adjacent  to  the  shorter 
side. 


Ex.  2.    Given  a  347,  b  642,  c  476,  to  find  the  angles. 

A  39°  11'  14^5. 
B  80°  43'  43".5. 
,  V  C  60°    5'    r. 

a- 


28  PLANE  TRIGONOMETRY. 

MISCELLANEOUS  EXAMPLES. 

1.  Given  A   45°  4',  B  75°  35',  c  457,  to  find  the  other 
parts. 

( C  59°  21'. 

Ans.   \  a  376.06. 

"  [h  514.48. 

2.  Given  a  454,  c  753,  A  45°  25',  to  find  the  other  parts. 

3.  Given  a  57,  6  89,  C  75°  4',  to  find  the  other  parts. 

/  A  36°  32'  37^. 
Ans.    \  B  68°  23'  23^ 
^.  (  c  92.495. 

f  4^  Given  a  41,  h  74,  c  63,  to  find  the  other  parts. 


(A  33°  37' 26^. 
Ans.    ^B88°    4' 12^ 
(  C  58°  18'  22^ 


6.   Given  h  75,  a  35,  to  find  the  other  parts. 

/A  27°  49'    b\ 
Ans.    ■]  B  62°  10'  55". 
( h  66.332. 
6.   Given  h  919,  A  37**  37',  to  find  the  other  parts. 

Ans    [^  560.94. 
I  h  727.95. 


7.  Given  h  45.3,  A  34°  23',  to  find  a  and  h. 

Ans    /  ^  30.998. 
^^'    \h  54.890. 

8.  Given  a  40,  6  57,  c  97,  to  find  the  other  parts. 

9.  Given  a  0.(55377,  h  0.06607,  A  45°,  to  find  the  other 
parts. 

/B60°  19'  34",  /B  119°  40' 26". 

Ans.    \  C  74°  40'  26",     or  J  C    15°  19'  34". 
(c  0.07334,  I  c  0.0201. 

10.  Given  a  54,  6  35,  B  97°  15',  to  find  the  other  parts. 


NOMETMC   FUNCTIONS. 


29 


CHAPTER  IV. 

GONOMETRIC   FUNCTIONS. 

ANALYTICAL  METHOD  * 

DEFINITIONS. 

57»  Instead  of  considering  the  Sine,  Tangent,  &c.  as  lines, 
having  a  certain  position  in  a  circle,  and  varying  not  only  as 
the  arc,  but  also  as  the  radius,  we  consider  them,  in  this  system, 
as  ratios,  varying  only  as  the  angle,  and  capable  of  being  repre- 
sented by  certain  lines  in  a  circle  only  when  the  radius  is 
unity. 

58.  The  Sine  of  an  angle  is  the  ratio  of 
the  side  opposite  it  in  a  right-angled  triangle 
to  the  hypothenuse. 

That  is,  if  in  any  right-angled  triangle 
A  B  C  we  represent  the  hypothenuse  by  A, 
and  the  sides  opposite  the  angles  A  and  B  by 
a  and  h  respectively, 


sm.  A  =  T 


sin.  B  = 


(1) 


59.    The  Tangent  of  an  angle  is  the  ratio  of  the  side  oppo- 
site it  in  a  right-angled  triangle  to  the  side  adjacent. 


That  is 


tan.  A  =  -r 
b 


tan.  B  =  - 
a 


(2) 


60.    The  Secant  of  an  angle  is  the  ratio  of  the  hypothenuse 
to  the  side  adjacent  to  the  angle. 


That  is 


sec.  A  =  7 

0 


sec.  B  = 


(3) 


61.    The  Cosine,  Cotangent,  Cosecant  of  an  angle  are  respec- 
tively the  sine,  tangent,  and  secant  of  its  complement. 

*  Those  who  have  taken  the  Geometrical  Method  can  omit  Chapters  IV. 
andV.  • 


30 


PLANE   TRIGONOMETRY. 


Therefore,  as  the  acute  angles  of  a  right-augled  triangle  are 
complements  of  each  other,  we  shall  have 

A  =  sin.  B  =  - 
h, 

B  r=  sin.  A  =  T 


COS. 


COS. 


cot.      A  =  tan.  B  = 


cot.      B  =:  tan.  A  = 


cosec.  A  =  sec.  B  =  - 
a 

cosec.  B  =  sec.  A  =  t 


(i) 


62.  By  inspecting  these  equations  it  will  be  seen  that  the 
sine  and  cosecant  of  an  angle  are  reciprocals  of  each  other ;  so 
also  the  cosine  and  secant,  and  the  tangent  and  cotangent. 
That  is 


sin. 

A  — 

cosec.  A 

cos. 

sec.  A 

tan 

A-       \ 

cot.  A 


or    i5osec.  A.=:  - 
or     sec.      A  = 
or     cot.      A  = 


sin.  A 

l__ 

COS.  A 
1 


tan.  A 


(5) 


63*    The  sine,  cosine,  dhc,  vary  only  as  the  angle  ;  that  is,  for 
a  given  angle  they  are  constant. 

Let  A  D  E  and  A  B  C  be  any  two  ^ 

right-angled  triangles,  having  a  com- 
mon angle  A ;  they  are  equiangular 
and  similar. 

Hence 


DE:DA  =  BC:BA,    or    ^  =  t^-;  = 


E  C 

DE_BC 
DA~"BA 

that  is,  the  sine  of  the  angle  A  is  constant,  whatever  the 
length  of  the  sides.  In  the  same  way  it  can  be  proved  that  the 
cosine,  tangent,  (fee.  of  a  given  angle  are  constant. 


TRIGONOMETRIC   FUNCTIONS. 


31 


64'«    Tlie  sine,  cosine,  <i:c,  can  he  represented  by  certain  lines  in 
a  circle,  when  the  radius  is  unity. 

Let  C  D  F  be  a  triangle,  right-angled 
at  F.  With  C  as  a  centre  and  C  D  as 
radius,  describe  a  circle  B  A  E  ;  pro- 
duce C  F  to  B,  and  draw  B I  parallel  to 
F  D,  and  meeting  C  D  produced ;  draw 
C  A  perpendicular,  and  D  H  and  A  K 
parallel  to  CB. 

In  the  right-angled  triangle  CDF 
DF 


A 

I 
K/ 

^^^ 

^ 

^ 

/ 

/ 

Ef 

/ 

>^ 

C         F      1 

sin. 


C  = 


DC 


cos. 

p        CF        HD 
CD~~  CD 

tan. 

DF        BI 
FC~~BC 

cot. 

p        CF        HD 
^  — FD~-HC" 

AK 
~  AC 

sec. 

CD_CI 
^""CF  —  CB 

cosec. 

CD        CD 
^~~DF~CH~ 

CK 
"CA 

If  CA,  CD,  CB 

that  is,  radius,  becomes  unity,  we  shall 

have 

sin.       C  =  D  F 
COS.       C  =  D  H 
tan.      C  =  B  I 
cot.      C  =  A  K 

' 

sec.       C  =  C  I 
cosec.   C  =  CK 

In  the  Geometrical  Method,  ivithout  limiting  the  radius  to 
unity,  these  lines  are  defined  as  the  sine,  cosine,  <fec.  of  the  arc 
or  angle  to  which  they  belong. 


65*    In  the  right-angled  triangle  ABC  (Art.  68) 

a^  +  62  =  h^ 


(6) 


32  PLANE  TRIGONOMETRY. 

From  (4)  *  and  (6)  we  have 


sin. 


.  *.  cos.^  A  =  1  —  sin.'^  A 


COS.  A  =  Va 
From  (4)  we  also  have 


sin.^  A 


(7) 
(8) 


sin.  A 
COS.  A 
sin.  A 
cos.  A 


=  -  z=  tan.  A 


tan.  A 


(9) 


If,  therefore,  the  sine  of  an  angle  is  known,  the  cosine  can  be 
found  by  (8),  the  tangent  by  (9),  then  the  cotangent,  the  secant, 
and  cosecant  by  (5).- 


66t    Problem.     To  find  the  sine  and  cosine  of  the  sum  and 
difference    of   two    angles,    when    their    sines    and    cosines    are 


Let  F  C  L  and  D  C  F  be  two  angles, 
represented  respectively  by  A  and  B. 

Draw  C  H,  so  as  to  make  the  angle 
F  C  H  =  D  C  F ;  then 

DCL  =  A  +  B 
HCL  =  A  — B 

From  the  points  D  and  H,  equally 
distant  from  C,  draw  D  I  and  H  L  per- 
pendicular to  C  L  ;  join  D  H  and  draw  F  K  perpendigular,  and 
F  E  and  H  G  parallel  to  C  L. 

The  triangles  C  D  F  and  C  F  H  are  equal. 

For  by  construction  C  D  and  C  H  and  the  angles  D  C  F  and 
F  C  H  are  equal,  and  C  F  is  common ;  therefore  D  F  is  equal 

*  In  the  Analytical  Method  these  numbers  standing  alone  in  parentheses 
refer  to  the  equations  with  the  same  number. 


TRIGONOMETRIC   FUNCTIONS.  33 

to  F  H,  and  D  H  is  perpendicular  to  C  F ;  and  the  triangles 
D  E  F  and  F  G  H  are  also  equal.  The  triangles  E  D  F  and 
F  C  Kj  having  their  sides  perpendicular  each  to  each,  are  bim- 
ilar ;  therefore  the  angle  EDF  =  FCK  =  A. 

Now 


.      ,,    ,    T.,       DI        FK  +  DE        FK    ,    DE 
sm.(A  +  B)=j^^=        DC        =DC+DC 

(10) 

Bin   (A       B)        ^^       FK-DE        FK        DE 

(11) 

Bin.  ^A       ±3;_jj^—        ^^        —DO       DC 

But 

FK        FK  ^  FC        FK  ^^  FC         .      ,          .    x 

dc==dc><fc""fcXdc==  '^"-  ^  "^'-  ^  1 

1 

.  (12) 

DE        DE  _  DF        DE  ,     DF                 »     .      -,. 

dc  =  dcXdf  =  dfXdc  =  ^^^-^^^^-^  J 

By  (12),  (10)  and  (11)  become 

sin.  (A  +  B)  =  sin.  A  cos.  B  +  cos.  A  sin.  B 

(13) 

sin.  (A  —  B)  =  sin.  A  cos.  B  —  cos.  A  sin.  B 

(14) 

Again, 

/A_LT^\       CI        CK  — EF        CK       EF  ,_, 

cos.(A  +  B)  =  ^-j^  =  -^^^    =CD-CD  (^^) 

/A         n\       CL        CK  +  EF        CK    ,    EF  ,_, 

C08.(A-B)^^=        J^        =-CD  +  cD  (^^) 


But 


CK  _  CK        CF  _  CK        CF  _ 
cd~cdXcf"~cf  ^CD"" ^^^'     ^^^' 

EF       EF  _  DF        EF        DF         .       .     .      ^ 
CD  =  CD  X  DF  =  DF  X  CD  =  ''"•  ^  '^^-  ^ 


(17) 


By  (17),  (15)  and  (16)  become 

cos.  (A  +  B)  =  COS.  A  COS.  B  —  sin.  A  sin.  B  (18) 

cos.  (A  —  B)  =  COS.  A  COS.  B  -\-  sin.  A  sin.  B  (19) 

a*  0 


34  PLANE   TRIGONOMETRY. 

We  can  write  (13)  and  (14)  in  one  formula  in  which  the 
upper  signs  correspond  to  each  other,  and  also  the  lower  ones ; 
so  also  (18)  and  (19),  as  follows : 

sin.  (A  ±  B)  =  sin.  A  cos.  B  ±  cos.  A  sin.  B  )  .^rvx 

cos.  (A  ±  B)  =  COS.  A  COS.  B  =F  sin.  A  sin.  B  / 


67  •  Theorem.  The  sum  of  the  sines  of  two  angles  is  to  the 
difference  of  their  sines  as  the  tangent  of  half  their  sum  is  to 
the  tangent  of  half  their  difference. 

The  sum  of  (13)  and  (14)  is 

sin.  (A  +  B)  +  sin.  (A  —  B)  =  2  sin.  A  cos.  B      (21) 

Their  difference  is 

sin.  (A  +  B)  —  sin.  (A  —  B)  =  2  cos.  A  sin.  B      (22) 

If  in  (21)  and  (22)  we  make 

A  +  B  =  M,  and  A  —  B  =  N 
that  is, 

A  =  J  (M  +  N),  and  B  =  J  (M  —  N) 

they  become 
sin.  M  +  sin.  N  =  2  sin.  i  (M  +  N)  cos.  i  (M  —  N)     (23) 
sin.  M  —  sin.  N  =  2  cos.  |(M  +  N)  sin.  J  (M  —  N)     (24) 

Dividing  (23)  by  (24),  we  have 

sin.  M  4-  sin.  K  _  sm.  ^  (M  +  N)  cos.  \  (M  —  N) 
sin.  M  —  sin.  N  ~  cos.  \  (M  -j-  N)  sin.  |  (M  —  N) 

Reducing  the  second  member  by  means  of  (9),  we  have 

sin.  M  -|-  sin.  IST tan.  ^  (M  -{-  ^)  /o5\ 

sin.  M  —  sin.  N  ""  tan.  ^  (M  —  N)  ^     ^ 

S8.  By  means  of  the  formulas  already  obtained,  the  sine,  co- 
sine, <fec.  of  angles  of  any  magnitude  can  be  found. 


TRIGONOMETRIC   FUNCTIONS.  35 

69.  To  find  the  sine,  Sc.  of  30°  and  60°. 

In  (13)  make  A  =  30°,  andB  =  30°;  as  30°  and  60°  are 
complements  of  each  other,  it  becomes 

sin.  60°  =  COS.  30°  ==  sm.  30°  cos.  30°  -f-  cos.  30°  sin.  30° 
=  2  sin.  30°  cos.  30° 

Dividing  bj  cos.  30",  we  have 

1  =  2  sin.  30° 
sin.  30°  =  J  =  cos.  60° 
whence  by  (5),  (8),  and  (9) 

cos.  30°  =  sin. 60°  =z y/l— sin.2  30°  =  yfl^  =  J V^3 

tai..  30°  =  cot.  60°  =  ^  =  ^3  =:  v/ J     ^  (gg^ 

cot.  30°  =  tan.  60°  =  ^7  =  V^3 

1  2 

sec.  30°  ==  cosec.  60°  =  z~.    =  — 
t  y  ^        Y^ 

cosec.  30°  =  sec.  60°  =  ^  =  2 

70.  To  find  the  sine,  dec.  of  45°. 

In  (7)  make  A  =  45°;  as  45°  is  complement  of  45°,  it 
becomes 
8in.2  45°  +  C0S.2  45°  =  2  sin.^  45°  =  2  cos.^  45°  r=  1 

sin.  45°  =  COS.  45°  =  y^J 
whence,  by  (5)  and  (9) 


tan.  45°  =  cot.  45°  =  ^^^^  =  1 
COS.  45° 

sec.  45°  =  cosec.  45°  = 7^  =  -—  z=J2 

COS.  45°        y  i        ^ 


(27) 


71.    To  find  the  sine,  dhc.  of  0°  and  90°. 

In  the  right-angled  triangle  ABC  (Art.  63)  let  A  B  revolve 
in  the  plane  ABC  about  A  as  a  centre  ;  when  the  angle  A  =  0, 
B  C  =  0,  and  A  C  =  A  B;  then 


(28) 


36  PLANE   TRIGONOMETRY. 

sin.  A  =  sin.  0°  =  cos.  90°  =  2-5  ==  q 

AB 

AC 

COS.  A  =  COS.  0°  ==  sin.  90°  =  t-^  =  1 

AB 

whence,  by  (5)  and  (9)  we  have 

no  X  nno  si".  0°  0  ^ 

tan.    0   =  cot.      90  = —  ^  -  =  0 

COS.  0°         1 

ano  X  Ao         sin.  90°         1 

tan.  90°  =  cot.        0°  = -^  =  -  =  oc 

cos.  90°        0 

sec.     0°  =  cosec.  90°  =  — ?-—  =  t  =  1 
cos.  0°         1 

sec.  90"  =  cosec.    0°  = ^rjr^  =  -  =  oo 

COS.  90°       0 


72.    To  find  the  sine,  <kc.  of  180°. 

In  (13)  and  (18)  make  A  =  90°,  and  B  r=  90°  ;  they  become 
by  means  of  (28) 

sin.  180°  =  sin.  90°  cos.  90°  +  cos.  90°  sin.  90°=  0 

COS.  180°  =  COS.  90°  COS.  90°  —  sin.  90°  sin.  90°=  —1 

whence,  by  (5)  and  (9), 

-  Q^o         sin.  180°  0  . 

tan.     180°  = --.^  =  — -  :=  0 

COS.  180°        — 1  . 

,       T  „  .,o        COS.  180°        —1  ^  r    (29) 

^^*-    1^^  =^hn8o^  =  -o-  =  ^' 

sec.      180°  = \--  =  -i-  =  ^1 

COS.  180°        — 1 


73.    To  find  the  sine,  <&;c.  of  270°. 

In  (13)  and  (18)  make  A  =  180°,   and  B  =  90° ;  they 
become  by  means  of  (28)  and  (29) 


TRIGONOMETRIC   FUNCTIONS.  37 

sin.  270°  =  sin.  180°  cos.  l?0°4-.cos.  180°  sin.  90°=— 1  ' 
COS.  270°  =  COS.  180°  cos.  90°  —  sin.  180°  sin.  90°  =  0 
whence,  from  (5)  and  (9) 


o^no        sin.  270°         —1 

tan.     270°  = —-^  =  — -  =  oo 

cos.  2/0°  0 


cot. 


COS.  270 


(30) 


74.  To  find  ike  sine,  d;c.  of  360°. 

In  (13)  and  (18)  make  A  =  180°,  and  B  =  180°;  they 
t»ecome  by  means  of  (29) 

sin.  360°  =  sin.  180° cos.  180°  +  cos.  180°  sin.  180°  =^  0  =  sin.  0°  )    ^i  \ 
cos. 360°  =  008.180° COS.  180°  — sin.  180° sin.  180°=l=cos.0°)  ^   ^ 

Hence  the  sine,  cosine,  &c.  of  360°  are  the  same  as  those  of  0°. 

75.  To  find  the  sine,  d;c.  of  any  angle. 

The  semi-circumference  of  a  circle  whose  radins  is  unity- 
is  3.14159265359.  If  we  divide  this  by  10800,  the  number  of 
minutes  in  the  semi-circumference,  it  will  give  the  length  of  the 
arc  of  r  =  0.000290888,  which  may  also  be  considered  the 
sine  of  an  angle  of  1'  (Arts.  64,  63,  and  Geom.,  III.  25). 

Hence  by  (8)  we  have 

cos.  V  =  V^l  —  sin.2  r  =  .9999999577 

Then  by  transposing  (21) 

sin.  (A  -[-  B)  =  2  sin.  A  cos.  B  —  sin.  (A  —  B) 
and  making  B  =  1',  and  A  =   1',  2',   3',  &c.  in  succession, 
we  obtain  for  the  sines 

sin.  2'  =  2  sin.  V  cos.  1'  —  sin.  0'  =  .0005817764 

sin.  3=2  sin.  2'  cos.  1'  —  sin.  V  =  .0008726646 

&c.  &c.  <fec. 


38  PLANE  TRIGONOMETRY. 

76.  Having  found  the  sines  up  t(J  45°,  the  cosines  up  to  45° 
can  be  found  by  (8)  : 

COS.  2'  =  v/l  —  sin.2  2'  =  .9999998308 
COS.  3'  =  v^l  —  sin.2  3'  =  .9999996193 

&C.  &C.  &c. 

77»  As  the  sine  of  an  angle  is  the  cosine  of  its  complement, 
the  sines  and  cosines  now  become  known  up  to  90°.  The 
tangents,  cotangents,  secants,  and  cosecants  can  now  be  found 
by  (5)  and  (9). 

78.  The  sines,  cosines,  &c.  of  all  angles  up  to  360°  are  now 
known ;  since,  disregarding  the  algebraic  signs,  they  are  the 
same  between  90°  and  180°,  180°  and  270°,  and  270°  and  360°, 
as  between  0°  and  90°.  This  will  become  evident  as  we  pro- 
ceed to  find  the  algebraic  signs  of  the  sines,  cosines,  &c. 

79.  Problem.  To  find  the  algebraic  signs  of  the  sines,  co- 
sines, (fee. 

For  angles  between  0°  and  90°. 

Since  the  trigonometric  functions  between  these  limits  are 
the  ratios  of  lines  which  we  assume  as  positive,  the  functions 
themselves  must  be  positive. 

80.  For  angles  between  90°  and  180°. 

In  (14)  and  (19)  make  A  =  180°,  and  B  <  90°;  they  be- 
come by  means  of  (29) 

sin.  (180°  — B)=:  sin.  1 80°  cos.  B  — cos.  180°  sin.  Bz=  sin.  B 
COS.  (180°  —  B)  =  COS.  180°  cos.  B  +  sin.  180°  sin.  B  =  —  cos.  B 

whence  by  (5)  and  (9) 

tan.  (180°  —  B)  =  —  tan.  B  cot.      (1 80°  _  B)  =  —  cot.  B 

sec.  (180°  _  B)  =  —  sec.  B         cosec.  (180°  —  B)  =  cosec.  B 


TRIGONOMETRIC  FUNCTIONS.  39 

That  is,  the  sine  and  cosecant  of  the  supplement  of  an  angle  are 
the  same  as  those  of  the  angle  itself ;  and  the  cosine^  tangent ^  co- 
tangent  J  and  secant  are  the  negative  of  those  of  the  angle. 

81.  For  angles  which  exceed  180°. 

In  (13)  and  (18)  make  A  =  180° ;  they  become  by  means 
of  (29) 

sin.  (180°  +  B)  =  sin.  180°  cos.  B  -fees.  180°  sin.  B  =  —  sin.  B 
COS.  (180°  +  B)  =cos.  180°  cos.  B  —  sin.  180°  sin.  Br=— cos.  B 

whence  by  (5)  and  (9) 

tan.  (1 80°  +  B)  =  tan.  B  cot.      (1 80°  +  B)  =  cot.  B 

sec.  (1 80°  -j-  B)  =  —  sec.  B       cosec.  (180°  +  B)  =  —  cosec.  B 

That  is,  the  tangent  and  cotangent  of  an  angle  which  exceeds 
180°  are  equal  to  those  of  its  excess  above  180°,  and  the  sine, 
cosine,  secant,  and  cosecant  of  this  angle  are  the  negative  of  those 
of  its  excess. 

82.  Corollary  1.  The  tangent  and  cotangent  of  angles  be- 
tween 180°  and  270°  are  the  same  as  for  angles  between  0°  and 
90°,  and  the  sine,  cosine,  secant,  and  cosecant  are  the  negative 
of  those  between  0°  and  90°. 

83.  Corollary  2.  The  cosine  and  secant  of  an  angle  between 
270°  and  360°  are  the  same  as  for  angles  between  0°  and  90°, 
and  the  sine,  tangent,  cotangent,  and  cosecant  are  the  negative 
of  those  between  0°  and  90°. 

84.  To  find  the  sine,  <fcc.  of  angles  which  exceed  360°. 

In  (13)  and  (18)  make  A  =  360°;  they  become  by  means 
of  (31) 
sin.  (360°  +  B)  =  sin.  360°  cos.  B  +  cos.  360°  sin.  B=  sin.  B 
cos.  (360°  4-  B)  =  cos.  360°  cos.  B  —  sin.  360°  sin.  B  =  cos.  B 

That  is,  the  sine,  cosine,  dx.  of  an  angle  which  exceeds  360° 
are  the  same  as  those  of  its  excess  above  360°. 


40  PLANE  TRIGONOI^ETRY. 

85 1    To  find  the  sine  and  cosine  of  double  a  given  angle. 
In  (13)  and  (18)  make  A  =  B,  and  they  become 
sin.  2  A  =  2  sin.  A  cos.  A 


COS.  2  A  =  cos.^  A 


A 


(32) 
(33) 


86.    To  find  the  sine  and  cosine  of  half  a  given  angle. 

Finding  the  sum  and  difference  of  (33)  and  (7),  we  have 
2  C0S.2  A  =  1  +  cos.  2  A,  and  2  sin.^  A  r=  1  —  cos.- 2  A     (34) 

In  (34)  substitute  |  A  for  A,  and  we  have 
2-  cos.2  J A^  1  +  cos.  A,  and  2  sin.^  i  A  =  1  —  cos.  A     (35) 

^  87.    From  the  results  obtained  in  the  preceding  articles  we 
form  the  following 

Table. 


0° 

1st  q. 

90° 

2dq. 

180° 

3dq. 

270° 

4th  q. 

(  n.  val. 
sin.  I    sign 
(  a.  val. 

0 
± 
±0 

incr. 

+ 
incr. 

1 

+ 
+1 

deer. 

+ 
deer. 

0 

± 
±0 

incr. 
.  deer. 

1 
—1 

deer, 
incr. 

(  n.  val. 
cos.  \    sign 
(  a.  val. 

1 

+ 
+1 

deer. 

+ 
deer. 

0 
± 
±0 

incr. 
deer. 

1 
—1 

deer, 
incr. 

0 
± 
±0 

incr. 

+ 

iner. 

{  n.  val. 
tan.  \    sign 
(  a.  val. 

0 

± 

±0 

incr. 

+ 

incr. 

00 

± 
±<« 

deer, 
iner. ' 

0 

± 

±0 

incr. 

+ 
in  r. 

oo 
± 

±00 

deer, 
incr. 

(  n.  val. 
cot.  ]    sign 
(  a.  val. 

00 

± 

±00 

deer. 

+ 
deer. 

0 

± 

±0 

incr. 
deer. 

00 

± 

±00 

deer. 

+ 
deer. 

0 
± 
±0 

incr. 
deer. 

(  n.  val. 
sec.  \    sign 
(  a.  val. 

1 

+ 
+1 

incr. 

+ 
incr. 

oo 
± 

deer, 
incr. 

1 
—1 

incr. 
deer. 

00 

± 

±00 

deer. 

+ 
deer. 

(  n.  val. 
cosec.  <    sign 
(  a.  val. 

oo 

± 

±00 

deer. 

+ 
deer. 

1 

+ 
+1 

incr. 

+ 
incr. 

00 

± 
±«) 

deer, 
incr. 

1 
—1 

iner. 
deer. 

q.,  quadrant.  n.  val.,  numerical  value.  a.  val.,  algebraical  value, 

incr.,  increasing.  deer.,  decreasing. 


SOLUTION   0.F  PEANE  TRIANGLES.  41 

r 

CHAPTER  V.^^ 

SOLUTION   OF   PLANE   TRIANGLES. 
ANALYTICAL  METHOD. 

88.  In  every  plane  triangle  there  are  six  parts,  three  sides 
and  three  angles.  Of  these,  any  three  being  given,  provided 
one  is  a  side,  the  others  can  be  found. 

RIGHT-ANGLED  TRIANGLES. 

89.  In  a  right-angled  triangle,  one  of  the  six  parts,  viz.  the 
right  angle,  is  always  given ;  and  if  one  of  the  acute  angles 
is  given,  the  other  is  known;  therefore,  in  a  right-angled 
triangle,  the  number  of  parts  to  be  considered  is  four,  any  two 
of  which  being  given,  the  others  can  be  found.  We  may  have 
four  cases,  according  as  there  are  given, 

1.  The  hypothenuse  and  an  acute  angle  ; 

2.  A  side  about  the  right  angle  and  an  acute  angle ; 

3.  A  side  about  the  right  angle  and  the  hypothenuse  ; 

4.  The  sides  about  the  right  angle. 

All  these  cases  can  be  solved  by  the  following  Theorem  : 

THEOREM  I. 

90.  /^  «^y  righi-angUd  plane  triangle^ 

1st.  The  side  opposite  an  acute  angle  is  equal  to  the  product  of 
the  sine  of  this  angle  and  the  hypothenuse. 

2d.  The  side  opposite  an  acute  angle  is  equal  to  the  jyroduct  of 
the  tangent  of  this  angle  and  the  side  adjacent  to  this  angle. 

*  Before  beginning  this  chapter,  the  arrangement  and  use  of  the  tables 
of  Logarithmic  Siues,  Cosiaes,  &c.  must  be  learned  from  Arts.  32,  33, 
and  34. 


42 


PLANE   TRlbONQMETKY. 


Let  ABC  (Art.  92)  be  a  triangle,  right-angled  at  C. 


By  (1)* 

sm.A  =  ^ 

-r»            ^ 

sm.  B  =  - 

h 

(36) 

hence 

a  =  A  sin.  A 

h  =zh  sin.  B 

(37) 

By  (2) 

tan.A^^ 

0 

tan.  B  =  - 
a 

(38) 

hence 

a  —  h  tan.  A 

b  =z  a  tan.  B 

(39) 

91  •    Corollari/.     As  sin.  A  =  cos.  B,  and  sin.  B  =  cos.  A 

a  =  k  cos.  B  b  =  h  COS.  A  (40) 


Case  I. 

92.    Given  the  hypothenuse   and  an  acute 
angle. 

By  (37)*  and  (40) 

a  =.h  sin.  A 
b  ■=^h  cos.  A 
Ex.  1.    Given  h  255,  and  A  57"  14',  to  find    a 
a  ai^d  6. 


By  Logarithms. 

a=zh  sin.  A  =  log.  255  -|-  log.  sin.  A 
h  255  ^  2.406540 

sin.  A  57°  14'  *  9.924735 

a  214.42  2.331275 

b  =  h  COS.  A  =  log.  255  -|-  log.  cos.  A 

A*  255  2.406540 

COS.  A  57°  14'  9.733373 

b  138.01  2.139913 

*  In  the  Analytical  Method  these  nuiphers  standing  alone  in  parentheses 
refer  to  the  equations  with  the  same  numher. 


SOLUTION   OF.   PLANE   TULAJ^GLES.  43 

93.  As  the  logarithmic  sine,  'cosine,  &c.  of  the  tables  are  in- 
creased by  10  (Art.  32),  the  resulting  logarithm  must  be  dimin- 
ished by  10  as  often  as  these  functions  appear  as  factors. 

94.  In  the  cases  that  follow,  where  division  is  performed  by 
logarithms,  we  add  the  complement.  As  the  logarithmic  sine, 
cosine,  &c.  of  the  tables  are  increased  by  10,  the  resulting  log- 
arithm is  the  logarithm  sought. 

Ex.  2.    Given  h  1676,  A  67°  13',  to  find  a  and  h. 

a  1545.23. 


Ans. 

6    649.03. 


■■{ 

Ex.  3.    Given  h  78.4,  B  15°  51',  to  find  a  and  h. 

{  a  75A2. 
\  h  21.41, 


Ans. 


Case  II. 

95.    Given  a  side  about  the  right  angle  and  an  acute  angle. 

By  (37) 

a  •=h  sin.  A 
b  z=  h  sin.  B 

Ex.  1.    Given  a  195,  B  64°  43',  to  find  b  and  h. 

a  195  2.290035 

sin.  A  =  cos.  64°  43'        comp.     0.369476 
A  456.57  2.659511 

h  2.659511 

sin.  B  64°  43'  9.956268 

b  412.84  2.615779 

Ex.  2.    Given  6  1075,  B  75°  49',  to  find  a  and  h. 

(a    271.68. 
^""    \  h  1108.79. 

Ex.  3.    Given  b  17.45,  A  47°  31',  to  find  a  and  h. 


Ans.    ^  ^  l^-^^- 
h  25.84. 


^{ 


44  PLANE  trIgonometry. 

Case  TII.  ' 

96  •    Given  a  side  about  the  right  angle  and 
the  hypothenuse. 
By  (36)  and  (37) 

sm.  A  z=  -- 


h  ^=  h  sin.  B 
Ex.  1.    Given  h  24.5,  a  17.4,  to  find  the  other  parts. 

a  17.4  1.240549 

h  24.5  comp.     8.610834 

sin.  A  45°  15' 5''  9.851383 

B  =  90°  —  45°  15'  5''  =  44°  44'  55'' 

^24.5  1.389166 

sin.  B  44°  44'  55"  9.847571 


h  17.248 

1.236737 

0th 

erwise  h  can 

be  found  from  the   formula  6^  =  7i^  —  a^. 

.  b=L\l{h^.a)  {h  —  a) 

Ex. 

2.    Given  h  172.8,  6  14.17,  to  find  the  other  parts. 

/A  85°  17' 47". 
Ans.   ^  B    4°  42'  13". 

1  a  172.218. 

Case  IV. 

97. 

Given  the  sides  about  the  right  angle. 

By  (38)  and  (37) 

tan.  A  =  7 

0 

^ 

h  =  " 

sin.  A 

Ex.  1.    Given  a  195,  6  147,  to  find  the  other  parts. 
a  195  2.290035 

b  147  comp.     7.832683 

tan.  A  52°  59'  22"  10.122718 


SOLUTION  OF  PLANE  TBIANGLES.  45 

B  =  90°  —  52°.  59;  2r  =  37°  0'  38'' 
a  195  '  2.290035 

sin.  A  52°  69'  22''  comp.    0.097712 

h  244.2  2.387747 

Otherwise  h  can  be  found  from  the  formula  h?'  =  a^  -|~  ^^  J 
then  the  angles  by  (36).  '^ 

Ex.  2.    Given  a  189,  6  14,  to  find  the  other  parts. 

r  A  85°  45'  49". 
Ans.    \  B    4°14'ir. 
1^189.518. 

98.    In  the  following  Examples  two  parts  of  a  right-angled 
triangle  are  given,  and  the  others  required. 

/  a  888.896. 
1.   Given  6  217,   A  915. 


Ans.    - 

A  76° 

16' 52". 

(b  13° 

43'   8". 

/A    5° 

d'  34". 

Ans.   - 

B84° 

50'  26". 

\h  1934.89. 

Ans.    i\ 
h 

64.038. 
66.503. 

2.  Given  a  174,   h  1927. 

3.  Given  a  17.94,   A  15°  39'. 

4.  Given  A  47.9,   A  59°  17'.  Ans.    /  «  ^l-^'^98. 

1  h  24.467. 

5.  Given  a  298,   h  744. 

6.  Given  a  9.75,   h  13.44. 

7.  Given  h  0.02518,   A  34°  T  10". 


/  A  23°  36'  42". 

Ans.    < 

B  66°  23'  18". 

I  h  681.712. 

( 

A  35°  57' 32". 

Ans.   < 

B  54°    2'  28". 

\ 

.  h  16.60. 

Ans 

1  a  0.01706. 
'*    \  h  0.03042. 

46 


PLANE  TRIGfONO^ETRY. 


OBLIQUE-ANGLED^  TRIANGLES. 

99.    In  solving  oblique-angled  triangles,  there  are  four  cases. 
There  may  be  given, 

1.  Two  angles  and  a  side  ; 

2.  Two  sides  and  an  angle  opposite  one  of  them ; 

3.  Two  sides  and  the  included  angle ; 

4.  The  three  sides. 

For  solving  these  we  demonstrate  the  three  following  Theorems. 


THEOREM  II. 

100«    In  any  'plane  triangle,  the  sides  have  the  same  ratio  as  the 
sines  of  the  opposite  angles. 

Let  a,  h,  c  represent  the  sides  op- 
posite the  angles  A,  B,  C,  respective- 
ly.    Then 

a  -.h  :  cz=  sin.  A  :  sin.  B  :  sin.  C      ""  h     T> 

From  B  draw  BD  perpendicular  to  h.     Then   ABD   and 
BDC  being  right-angled  triangles,  by  (37)  we  have 

c  sin.  A  =  B  D  =  a  sin  C 
hence  a  :  c  =z  sin.  A  :  sin.  C 

In  like  manner  it  can  be  shown  that 

a  :  b  =  sin.  A  :  sin.  B 

b  :  c  =  sin.  B  :  sin.  C 
Therefore  a  :  b  :  c  =  sin.  A  :  sin.  B  :  sin.  C 

101.  If  the  perpendicular  falls  without 
the  triangle,  the  angles  B C  A  and  BCD,  be- 
ing supplements  of  each  other,  have  the  same 
sine,  and 

B  D  =  a  sin.  B  C  D  =  a  sin.  C 


SOLUTION   OF  PLA'NE  TRIAJ^GLES. 


47 


THEOREM.ni. 

102.  In  any  plane  triangle,  the  sum  of  any  two  sides  is  to  their 
differ ence,  as  the  tangent  of  half  the  sum  of  the  opposite  angles  is 
to  the  tangent  of  half  their  difference. 

Let  ABC  (Art.  103)  be  a  plane  triangle ;  then 

a  -\-  c  :  a  —  c  =  tan.  J  (A  +  C)  :  tan.  J  (A  —  C) 
By  (41)  we  have 

a  \  c  •=:  sin.  A  :  sin.  C 

By  composition  and  division  (Geom.  Pn.  19) 
a  -\-  c  :  a  —  c=z  sin.  A  -\-  sin.  C 

a  4-  c        sin.  A  4-  sin.  C 

or  — ' —  = ' 

a  —  c        sin.  A  —  sin.  C 

But  by  (25) 

sin.  A  -f-  sin.  C tan.  ^  (A  -}-  C) 

sin.  A  —  sin.  C        tan.  ^  (A  —  C) 

a  -\-  c tan.  ^  (A  -f-  C) 

C) 

or      a  -\-  c  :  a 


sin.  A  —  sin.  C 


Therefore 


a  —  c        tan.  ^  (A 
b  =  tan.  1  (A  4-  C)  :  tan.  J  (A  —  C)     (42) 


THEOREM    IV. 

103*  In  any  plane  triangle,  the  cosine  of  any  angle  is  equal  to 
the  sum  of  the  squares  of  the  two  adjacent  sides  minus  the  square 
of  the  opposite  side,  divided  by  twice  tJie  product  of  the  adjacent 
sides. 

In  the  triangle  ABC 

CD  =  5  — AD 
CD2  =  52  —  25AD  +  AD2 
Adding  BD^  to  both  members,  we  have 
(Geom.,  II.  27) 

But  by  (40)  A  D  =  c  cos,  A 

.-.  a^  =  6^*  -|-  (T*  —  2  6c  cog.  A 

—  6'  -f  c*  —  q* 
2  6c 


Therefore 


A  = 


(43) 


48 


PLANE  TRiaONOMETRY. 


104.  For  greater  convenience  infusing  logarithms  (43)  can  be 
changed  by  subtracting  both' members  from  unity  and  reducing, 
as  follows : 


1  —  cos.  A 


2hc—y'—(?^a^__o}—{h  —  cf 
2bc  ~~  2bc 

(a  —  h  -{-  c)  (a  -{-  b  —  c) 


But  by  (35) 


~~  2  6c 

1  —  cos.  A  =  2  sin.^  J  A 


(44) 


Substituting  in  (44)  this  value  of  1  —  cos.  A,  and  also  put- 


tmg  8 


-,  and  reducing,  we  have 


In  like  manner 


sin.  J  A  =  4/ 


sm. 


sm.  h  V  = 


iC  =  y/. 


c^- 

-b){s- 

-c)] 

bc 

'(.9- 

-a)(s- 

-c) 

ac 

c^- 

-a)(s- 

-b) 

ab 

(45) 


105»    Adding  both  members  of  (43)  to  unity,  we  have 


1  -|-  COS.  A 


__  2bc-\-b^-\-(^  —  a^  __  (b  -}-  cy  —  a* 


2b 


2  be 


__  (6  -f  c  +  g)  (6  -f  c  —  g) 


26c 


(46) 


But  by  (35)  1  +  cos.  A  =  2  cos.^  J  A 


Substituting  in  (46)  this  value  of  1  -|-  cos.  A,  and  the  value 
of  s  as  in  Art.  104,  and  reducing,  we  have 


In  like  manner 


cos. 


cos. 


cos. 


s  (s  —  g) 
b~c 

,9  (s  —  b) 


c) 


(47) 


SOLUTION  «JF  PLANE  TtllANGLES. 


49 


106.    Dividing  equations  (45)  \)j  (47)  in  order,  by  means 
of  (9)  we  have 


^  Y       s  (s  —  a) 

2  y       s  (s  —  b) 

tan.iC^.A^-;^^^-^> 
^         y     s  (s  —  c) 


(48) 


Case  I. 

107*    (riven  two  angles  and  a  side. 
By  (41) 

sin.  A  :  sin.  B  :  sin.  C  =  a  :  b  :  c 

Ex.  1.    Given  A  48°,  C  55°  17',  a  417,  to 
find  the  other  parts.  ° 

B  =  180°  —  (55°  17'  +  48°)  =  76°  43' 


sin.  A  48° 
:  sin.  B  76°  43' 
=  a417 
:  b  546.12 

sin.  A  48° 
:  sin.  C  55°  17' 
=  a417 
:  c  461.24 


comp.  0.128927 
9.988223 
2.620136 


2.737286 

comp.  0.128927 
9.914860 
2.620136 
2.663923 


Ex.  2.   Given  A  95°  4',  B  25°  14',  c  49.17,  to  find  the  other 
parts. 

C  59°  42'. 

Ans.    ^  a  56.727. 

b  24.278. 


50  PLAlifE   TRIGONOMETRY. 

Case   II. 

108i    Given  two  sides  and  an  angle  opposite  to  one  of  them. 

By  (41) 

a  :h  :  €■=  sin.  A  :  sin.  B  :  sin.  C 

Ex.  1.    Given  a  55,  c  49.87,  A  25°  44',  to  find  the  other  parta 

a  55  comp.  8.259637 

:  c  49.87  1.697839 

=  sin.  A  25°  44'  9.637673 

:  sin.  C  23°  ir  2"'  9.595149 

B  =  180°  —  (23°  11'  r  4-  25°  44')  =  131°  4'  58" 

sin.  C    23°  11'    r  comp.  0.404851 

:  sin.  B  131°    4' 58^'  9.877234 

=  c  49.87  1.697839 

:  h  95.483  1.979924 

109.  If  B  C,  the  side  opposite  the 
given  angle,  is  less  than  the  other 
given  side  AB,  and  the  given  angle  A- 
is  acute,  there  are  two  triangles  which 
satisfy  the  conditions,  viz.  ABC  and  A  B  D,  in  which  the 
angles  B  C  A  and  B  D  A  are  supplements  of  each  other.  The 
Log.  sine  obtained  in  working  such  an  example  represents 
either  the  angle  BCA,  or  its  supplement  BDA  (Art.  80).  If 
the  given  angle  A  is  obtuse,  or  the  side  opposite  the  given 
angle  is  greater  than  the  other  given  side,  there  is  but  one 
solution  (Geom.,  VI.  11).  Whenever  the  solution  is  impossi- 
ble (Geom.,  VI.  12),  the  Log.  sine  obtained  in  working  the  ex- 
ample will  be  greater  than  unity  (10.  in  the  tables),  which 
is  impossible  (Art.  58). 


,-''C 


SOLUTION  *0F  PLANE  TRIANGLES.  51 


Ex.  2.    Given  a  95.5,  c  173.2,  A  27°  4',  to  find  the  other 
parts. 

/  C  55°  36'  47'',  (  C  124°  23'  13''. 

Ans.   \  B  97°  19'  13",     or  <|  B    28°  32'  47". 
(6  208.17,  16  100.29. 


Case  III. 

llOt    Given  two  sides  and  the  included  angle. 

By  (42)    . 

a  -{-  c  :  a  —  c  =:  tan.  J  (A  -|-  C)  :  tan.  J  (A  —  C)  < 

By  (41) 

sin.  A  :  sin.  B  :  sin.  C  =:  a  :  h  :  c 

Ex.  1.   Given  a  976,  c  89,  B  51°  17',  to  find  the  other  parts. 

J  (A  +  C)  =  J  (180°  —  51°  17')  =  64°  21'  30"" 

a-\-c=  1065  comp.  6.972650 

:  a  _  c  =  887  •         2.947924 

=  tan.  |(A  +  C)  —  tan.  64°  21'  30"  10.318746 

:  tan.  i  (A  —  C)  i=  tan.  60°    2'  36"  10.239320 

Half  the  sum  plus  half  the  difference  gives  the  greater  angle 
A  124°  24'  6";  half  the  sum  minus  half  the  difference,  the 
less  C  4°  18'  54". 

sin.  C    4°  18'  54"  comp.  1.123553 

:  sin.  B  51°  17'  9.892233 

=  c  89  1.949390 

:  6  922.94  2.965176 

Ex.  2.  Given  a  91,  6  104,  C  14°  30',  to  find  the  other 
parts. 

/A    55°    5' 37". 
Ans.    ^B  110°  24' 23". 
I  c  27.783. 


52 


PLANE  TRlGONOIikTKY. 


Case  IV. 
lilt    Given  the  three  sides. 
By  (48) 


tan.  J  A  =  4 /- 


tan, 


_     l{s-h){s-c) 
(s  —  a) 

-a)  {s  —  c) 


s{s  —  b) 
Ex.  1.    Given  a  125,  b  135,  c  75,  to  find  the  angles. 


s  167.5 

comp.  7.775985 

comp.  7.775985 

comp.  7.775985 

s  —  a    42.5 

comp.  8.371611 

1.628389 

1.628389 

s  —  b    32.5 

1.511883 

comp.  8.488117 

1.511883 

s—c    92.5 

1.966142 

1.966142 

comp.  8.033858 

2)19.6256^1 

2)19.858633 

2)18.950115 

Log.  tangents          9.8128105 

9.9293165 

9.4750575 

^  A  33°  1-3.6" 

i  B  40°  21' 28.3" 

^  C 16°  37'  28.1" 

A  66°  2' 7.2" 

B  80°  42' 56.6" 

C  33°  14' 56.2" 

112.  The  angles  can  also  be  obtained  from  formula  (45)  or 
(47).  As  the  sines  differ  from  each  other  more  for  angles  be- 
tween 0°  and  45°,  the  cosines  for  angles  between  45°  and  90°, 
(45)  is  preferable  when  the  half-angle  is  less  than  45°,  and  (47) 
when  the  half-angle  is  more  than  45°.  But  (48)  is  more  accu- 
rate for  aU  angles,  and  requires  but  four  logarithms. 

113*  The  sum  of  any  two  sides  must  be  greater-  than  the 
remaining  side,  otherwise  the  triangle  is  impossible. 


Ex.  2.    Given  a  347,  b  542,  c  476,  to  find  the  angles. 

(A39°  ir  14''.5. 
Ans.  \  B  80°  43'  43'^5. 
( C  60°    5'    r. 


For  Miscellaneous  Examples  see  page  28. 


PEACTieAL  APPLICATIONS.  53 

CHAPTER  VI. 

PRACTICAL   APPLICATIONS. 
DEFINITIONS. 

Ill*  A  Horizontal  Plane  is  a  plane  which  is  tangent  to  the 
earth's  surface,  and  every  line  in  this  plane  is  a  horiz(mtal  line. 

115*  A  Vertical  Line  is  a  line  which  is  perpendicular  to  a 
horizontal  plane,  and  every  plane  including  in  its  surface  such  a 
line  is  a  vertical  plane. 

116.  A  Horizontal  Angle  is  one  that  has  the  plane  of  its 
sides  horizontal. 

117.  A  Vertical  Angle  is  one  that  has  the  plane  of  its  sides 
vertical. 

118.  An  Angle  of  Elevation  is  a  vertical  angle  having  one  side 
horizontal,  and  the  inclined  side  above  it ;  as  C  AB  (Art.  120). 

119.  An  Angle  of  Depression  is  a  vertical  angle  having  one 
side  horizontal,  and  the  inclined  side  below  it ;  as  F  B  A. 

HEIGHTS  AND  DISTANCES. 
PROBLEM  I. 

120.  To  determine  the  height  of  a  vertical  object  standing  on  a 
horizontal  plane. . 

Suppose  it  is  required  to  find  the  height 
of  BC. 

From  the  foot  of  B  C  measure  any  con- 
venient distance  CA,  and  at  A  take  the 
angle  of  elevation  CAB.  Then,  in  the 
right-angled  triangle  ABC,  all  the  angles 
and  the  side  AC  are  known,  and  BC  can  be  found 


D  C 


54  PLANE  TK|Gt)NOIl!jETRY. 

121  •  Secoiid  Method.  WitLout  ineasur- 
ing  the  angle  of  elevation  C  A  B,  -the  'height 
of  B  C  may  be  found,  as  follows  : 

Cut  a  stake  equal  in  length  to  the  dis- 
tance of  your  eye  from  the  ground  ;  move 
away  from  B  C,  until,  by  taking  the  position 
A  D,  with  your  head  at  A  and  the  stake  D  E  standing  perpen- 
dicular at  your  feet,  you  can  just  see,  in  a  line  with  the  top  of 
the  stake,  the  top  of  the  object. 

Then,  since  AD  is  equal  to  DE,  AC  is  equal  to  CB 
(Geom.  II.  20) ;  and  if  A  C  is  measured,  C  B  becomes  known. 

This  method  is  used  in  finding  the  heights  of  trees,  or  the 
length  of  that  part  of  the  tree  which  is  fit  for  timber. 

122 1  Third  Method.  When  shadows  are  cast  by  objects,  still 
another  method  can  be  used. 

Measure  the  height  of  any  convenient  vertical  object  and  the 
length  of  its  shadow,  and  also  the  length  of  the  shadow  of  the 
object  whose  height  is  sought.  Then  (Geom.  II.  20)  the  length 
of  the  shadow  of  the  one  is  to  its  height  as  the  length  of  the 
shadow  of  the  other  to  its  height. 

123i  If  the  height  B  C  is  known,  by  taking  any  position,  as 
A,  and  measuring  the  angle  BAC,  the  distance  AC  can  be 
found. 

124.  If  from  the  top  of  the  tower  the  angle  of  depressio-i 
F  B  A,  which  is  equal  to  BAC,  is  taken,  then,  if  B  C  is  known, 
A  C  can  be  found  ;  if  A  C  is  known,  B  C  can  be  found. 

Ex.  1.  The  distance  A  C  is  100  feet,  and  the  angle  BAC  41° 
15' ;  what  is  the  height  of  the  tower  1  Aus.  87.6976  ft. 


PROBLEM  II. 

125.    To  find  the  height  of  a  vertical  object  standing  on  an  in- 
clined plane. 


PRACflCAI    APPLICATIONS. 


55 


Measure  any  convenient 
distance  from  the  object, 
as  C  D,  and  the  angles  of 
elevation  of  both  D  C  and 
D  B.  Then  all  the  angles, 
and  the  side  D  C,  of  the 
triangle  D  B  C,  are  known, 
and  A  C  can  be  found. 


Ex.  1.    If  DC  is  55  feet,  and  the  angle  of  elevation  of  DC 
is  10°,  and  of  D  B  36°,  whal  is  the  height  of  the  tree  1 
Angle  B  D  C  =  36°  —  10°  =  26° 
Angle  D  B  C  =  90°  —  36°  ==  54° 

Ans.  29.8  ft. 

126«  Second  Method.  If  two  stations  be  taken  in  the  line 
AC,  as  A  and  D,  and  the  distance  A D  and  the  angles  of  eleva- 
tion of  A  B,  D  B,  and  D  C  are  measured,  then  in  the  triangle 
A  B  D  aU  the  angles  and  the  side  A  D  are  known,  and  B  D  can 
be  found ;  then  in  the  triangle  B  D  C  all  the  angles  and  the 
side  B  D  are  known,  and  B  C  and  C  D  can  be  found. 

Ex.  2.    If  A  D  is  11.5  feet,  and  the  angle  of  elevation  of  A  B 
38°,  of  D  B  41°,  and  of  D  C  9°,  what  is  the  height  of  the  tree  1 
Angle  BAD  =  29°,  BDC  =  32°,  ABD=:3°,  BCD=f:99°. 

\   ^^       /'^j  .■?■•  Ans.  57.J^6  ft. 


y> 


/  ij  ^ 


PROBLEM   III. 


127.    to  find  the  height  of  an  inaccessible  object  above  a  hori- 
zontal plane. 

The  first  method  is  pre- 
cisely the  same  as  the 
last  method  in  Prob.  II., 
the  angle  of  elevation  of 
A  D  being  0°. 


56  PLANE  TRIGONOME^Y. 

128.  Second  Method.  Rule.  Divide  the  distance  between  the 
stations  by  the  difference  of  the  natural  cotangents  of  the  angles  of 
elevation. 

Demonstration.* 
With  B  as  the  centre,  and 
B  A  as  radius,  describe  the 
arc  AE;  produce  BC  till  it 
meets  the  arc  at  E;  at  the 
point  E  draw  E  G  tangent  to 
the  arc,  and  produce  B  D  and 
BA  toH  andG. 

AD  :GH  =  BD 
AD  :GH=:BC 


BC  = 


BE  X  AD 
GH 


But  B  E  is  radius  or  unity,  and  G  H  is  the  difference  between 
the  tangents  of  the  angles  G  B  E  and  H  B  E,  that  is,  between 
the,  cotangents  of  B  A  C  and  B  D  C  : 

AD 


.-.  BC 


cot.  BAG  — cot.  BDC 


Ex.  1.  If  the  distance  AD  is  97  feet,  and  the  angles  of  ele- 
vation of  AB  and  DB  are  respectively  37°  22'  and  56°  10', 
what  is  the  height  B  P  ] 

t  cot.  37°  22'  1.30952 

cot.  56°  10'  0.67028 


129.    Third  Method. 


0.63924)97.00000(151.7  ft.  Ans. 
Measure  any  base  line  AD,  and  the 


'i^  Analytical.     AC  =  BCtan.  ABC  DC  =  BC  tan.  DBC 

.-.  AC— DC  =  AD  =  BC(tan.  ABC  — tan.  DBC) 

BC  = 


AD 
cot.  BAG  —  cot.  BDC 


+  To  find  the  Nat.  Cot.  from  the  Log.  Tables  aubtract  10  from  the  char- 
acteristic of  the  Log.  Cot.,  and  then  find  its  corresponding  natural  number 
from  the  Table  of  Logarithms. 


PRACTICAL  APPLICATIONS. 


57 


horizontal  angles  CAD  and 
CD  A,  and  the  vertical  an- 
gle CAB.  Then,  in  the 
triangle  A  CD,  we  have  one 
side  and  all  the  angles,  and 
A  C  can  be  found  ;  then,  in 
the  right-angled  triangle 
ABC,  we  have  one  side  and 
all  the  angles,  and  the  height  B  C  can  be  found. 

Ex.  2.  At  the  point  A,  I  took  the  angle  of  elevation  of  the 
top  of  the  tower  B  C  34°  45' ;  then,  turning  at  a  right  angle,  I 
measured  off  A  D  40  feet,  and  measured  the  angle  ADC  58°. 
What  is  the  height  of  the  tower  1  Ans.  44.4  ft. 

PROBLEM  lY. 
ISOi    To  find  the  distance  of  an  inaccessible  object. 

Measure  a  horizontal  base  line  A  C, 
and  the  angles  BAC  and  ACB.  Then, 
in  the  triangle  ABC,  we  have  one  side 
and  all  the  angles,  and  A  B  or  C  B  can 
be  found. 

Ex.  1.    If  A  C  is  20  chains,  the  angle 


A  25°, 
AB1 


and  C  92°,  what  is  the  distance 
Ans.  22.43  chs. 

PROBLEM  V. 


131  •    To  find  the  distance  between  two  objects  separated  by  an 
impassable  barrier. 

Take  any  station  C,  from  which  both  A 
and  B  are  visible  and  accessible.  Measure 
the  angle  ACB,  and  the  sides  AC  and 
CB.  Then,  in  the  triangle  ABC,  we  have 
two  sides  and  the  included  angle,  to  find 
the  third  side  A  B. 


58 


PLANE  trigonometry: 


132t  If  the  point  A  can  be  seen  at  B,  the  angle  ABC  can 
also  be  measured,  and  then  only  one  side,  A  C  or  C  B,  whichever 
is  most  convenient,  need  be  measured.  Then,  in  the  triangle 
ABC,  we  have  one  side  and  all  the  angles,, and  A  B  can  be  found. 

Ex.  1.  If  AC  is  44.4  chains,  C  B  50  chains,  and  the  angle  C 
39°  25',  what  is  the  distance  AB1  Ans.  32.268  chs. 

133i  If  there  is  an  elevated  object,  whose  height  is  known, 
in  the  line  A  B  produced,  from  which  the  points  A  and  B  can 
both  be  seen,  by  taking  the  angles  of  depression  of  A  and  B, 
the  distance  A  B  can  be  found  by  a  rule  the  reverse  of  that 
given  in  Art.  128,  viz.  :  Multiply  the  difference  of  the  natural 
cotangents  of  the  angles  of  depression  hy  the  height  of  tlie  object. 

Ex.  2.  Wishing  to  know  the  width  of  a  river,  from  the  top 
of  a  tower  197  feet  above  the  level  of  the  river,  I  found  the 
angle  of  depression  of  the  nearer  edge  54"  10',  of  the  farther 
48°  37'  j  what  was  the  width  of  the  river  ]         Ans.  31.3  ft. 


PROBLEM  VI. 
131*    To  find  the  distance  between  two  inaccessible  objects. 

Measure  any  convenient  base 
line  AD,  and  the  angles  BAD, 
BDA,  CDA,  and  CAD. 

Then,  in  the  triangle  ABD, 
we  have  all  the  angles  and  the 
side  A  D,  and  B  D  can  be  found. 
In  the  triangle  ACD  we  have  all 
the  angles  and  the  side  A  D,  and 
C D  can  be  found.  Then,  in  the  triangle  BCD,  the  two  sides 
B  D  and  D  C  and  the  included  angle  are  known,  and  B  C,  the 
distance  required,  can  be  ^-^"^^ 

Ex.  1.  If  Ai>-ir40"roHriong,  the  angle  BAD  100°,  ADB 
51°,  CDA  120°,  and  CAD  55°,  what  is  the  distance  between 
BandC]  Ans.  355.05  rds. 


(q^  (H^U  C 


lATIONS.  59 

PROBLEM  VII. 

135*  To  find  the  distances,  from  a  given  pointy  of  three  objects 
whose  distances  from  each  other  are  known. 

Let  D  be  the  given  point,  and  A,  B, 
C  three  points  whose  distances  from  each 
other  are  known;  it  is  required  to  find 
the  distance  from  D  to  the  several  points. 
The  angles  ADB  and  BDC  must  be 
measured.  Then  describe  a  circumfer- 
ence through  the  three  points  A,  D,  C; 
draw  AB,  BC,  AC,  AD,  BD,CD;  from 
A  and  C  draw  lines  to  E,  the  point  where 
B  D  cuts  the  circumference.  In  the  triangle  A  E  C  the  side  A  C 
is  given,  and  all  the  angles  are  known ;  for  E  C  A  =  E  D  A, 
and  CAE=  CDE  (Geom.,  IIL  22);  therefore  AE  can  be 
found. 

In  the  triangle  ABC,  the  three  sides  being  given,  the  three 
angles  can  be  found.  Then,  in  the  triangle  ABE,  we  know  the 
sides  AB,  AE,  and  the  included  angle  B AE  (=  BAC  —  EAC), 
and  the  angle  ABE  can  be  found.  Then,  in  the  triangle  ABD, 
all  the  angles  become  known,  and  the  side  AB  is  given ;  there- 
fore AD  and  BD  can  be  found ;  then  C  D  can  also  be  found. 

13S.  If  the  point  B  is  between  D  and  the  line  A  C,  the  angle 
B  A  E  =  B  A  C  4-  E  A  C.  But  in  this  case  the  distances  A  B 
and  B  C  cannot  be  the  same  as  when  B  is  beyond  the  line  A  C, 
unless  B  D  cuts  A  C  at  right  angles. 

If,  however,  B  D  cuts  A  C  at  right  angles,  and  the  position  of 
B  is  not  known,  though  the  distances  of  A  and  C  from  D  can 
be  found,  the  distance  of  B  will  be  ambiguous ;  B  may  be  in 
either  of  two  points  in  the  line  B  D. 

If  the  angle  B  is  the  supplement  of  ADC,  the  point  B  will 
fall  on  E,  and  BCA  =BDA  and  CAB  =  CDB.  In  this 
case  D  may  be  anywhere  in  the  arc  ADC,  and  the  distances 


60  PLANE  TRIGONOMETRY. 

A  D,  B  D,  C  D  cannot  be  determined  from  the  data  given.    Also, 
if  A,  B,  C,  D  are  in  the  same  straight  hne,  the  distances  cannot 

be  determined.  ^ ,^  x 

•Ex.  1.   If  AB  (Fig.  Art.l35)   is   50,   B  C  65,  AC   38.62        > 
chains  in  length,  the  angle  ADB  10°,  BD  C  12°  U',  what  are     '  / 
the  distances  A  D,  B  D,  C  D  1       ^J        '  '^^'-       -d:!:     -   ^^ 
Ans.  AD  100,  BD  145.37,  CD  84.828  chs. 


DETERMINATION  OF  AREAS. 

137»  The  Areas  of  triangles,  parallelograms,  and  trapezoids, 
when  their  altitudes  are  given,  can  be  found  bj  application  of 
the  principles  already  demonstrated  in  Geometry.  But  by 
Trigonometry  the  areas  of  these  polygons  can  be  found  when 
in  the  triangle  and  parallelogram,  in  place  of  the  base  and  alti- 
tude, two  adjacent  sides  and  an  angle,  and  in  the  trapezoid  the 
sides  and  two  opposite  angles,  are  given. 

PROBLEM  I. 

138.  To  fifid  the  area  of  a  parallelogram.  "--^ 

Rule  I.    Multiply  the  base  hy  the  altitude.     (Geom.,  II.  10.) 
Ex.  1.    How  many  square  yards  are  there  in  the  sides,  floor, 

and  ceiling  of  a  rectangular  room,  20  feet  long,  16  feet  wide, 

and  10  feet  high] 

139.  If  two  adjacent  sides  and  an  angle  are  given,  the  area 
can  be  found  by  . 

Rule  II.  Multiply  together  the  two  adjacent  sides  and  the  sine 
of  the  included  angle. 

For,  by  Theorem  I., 

ED  =  ADXsin.A  A_ 

Area  =  AB  X  ED  =  AB  X  AD  X  sin.  A 

If  the  work  is  done  by  logarithms,  ten 
must  be  taken  from  the  index  of  the  log. 
sine.     (Art.  32.) 


PKAOTICAL  APPLICATIONS.  Gl 

Ex.  2.   What  is  the  area  of  a  parallelogram  whose  adjacent 
sides  are  475  and  355  feet,  and  the  included  angle  49°  ?  '^\  7  7/^ 


PROBLEM  II. 

HO.    To  find  the  area  of  a  triangle. 

Rule  I.  Multiply  one  half  the  base  hy  the  altitude,  (Geom., 
11.  11.) 

141.  As  a  triangle  is  half  a  parallelogram  of  the  same  base 
and  altitude,  when  two  sides  and  the  included  angle  are  given, 
the  area  can  be  found  (139)  by 

Rule  II.  Multiply  together  tJie  two  sides  and  half  the  sine  of 
the  included  angle. 

Ex.  1.  What  is  the  area  of  a  triangle  whose  two  sides  are  76 
and  14  rods,  and4:he  included  angle  71°  ir  f,  f/^f^tc  /^ 

142*  When  the  three  sides  are  gi^en,  an  angle  can  be  found, 
and  then  the  area  by  the  last  rule ;  or,  without  finding  an  angle, 
the  area  can  be  found  by  the  following  rule  : 

Rule  III.  From  half  the  sum  of  the  three  sides  subtract  succes- 
sively the  three  sides  ;  mmdtiply  together  these  three  remainders  and 
the  half-sum,  and  extract  the  square  root  of  the  product. 

Let  a,  b,  c  denote   respectively  the 
sides  opposite  the  angles  A,  B,  C. 
DC  =:AC  — AD 
DC2  =  AC2— 2  ACX  AD  +  AD^     A^-^ ^ ^C 

Adding  B  D^  to  both  members,  by  Geom.,  II.  27,  we  have 
BC2  =  AC2  +  AB2  — 2  ACX  AD 
or  ^2  _  52  _j_  ^2  _  2  5  X  AD 

b^±^-^ 


62  PLANE  TRIGONOMETRY. 

But  (Geom.,  II.  28)      B  D^  =  A  B^  —  A  D^ 


AC  X  BD         h      lWc^~{c^-\-a^  —  hy 


2         ~~2y  4&^ 

^4  62  c«  —  (&2  -I-  c«  —  g^)^ 
16 


As  the  product  of  the  sum  and  difference  of  two  quantities 
is  equal  to  the  difference  of  their  squares,  we  have 
4  h^^—  {V^J^c^-a'f=(^  &c  — [6^+c^— a2J)X(2  &c+[6^+c^— a^j) 

But 

2&C  — (S'^  +  c*  — a«)  =  a«  — (&"— 2  6c  +  c«)=:a2— (&  — c)« 

and 

«8_(6_c)2=(a4-[&  — c])  X  (a  — [&  — c])  =  (a+&— c)  X  (a+c— &) 

and  so  also 

s/ 

a  -\-  h  -\-  c 


(a-f-J-f  c)  X  (6+c  — a)  X  (g-j-c  — &)  X  («+ ^  — ^ 
.-.  Area  =  i  / ^q 


Putting  i 

we  have  Area  =z^s  (s  —  a)  {s  —  b)  {s  —  c) 

Ex.  1.    What  is  the  area  of  a  triangle  whose  sides  are  45,  55, 
and  60  feet  1 

5  =  ^  =  80  Log.      1.903090 
80  —  45  =  35    "  1.544068 

80  —  55  =  25    "  1.397940 

80  —  60  =  20   "  1.301030 

2)6.146128 
3.073064 

Ans.  1183.2  sq.ft. 


PRACTICAL  APPLICATIONS.  63 

PROBLEM  III. 

143«    To  find  the  area  of  a  trapezoid. 

Rule  I.  Multiply  half  the  sum  of  the  parallel  sides  by  the  per- 
pendicular distance  between  them. 

144*   If  the  angles  are  known,  we  can  use 

Rule  II.  Divide  the  trapezoid  by  a  diagonal^  and  find  the  area 
of  each  triangle  (141)  ;  their  sum  mill  be  the  area  required. 

145.  If  the  length  of  the  diagonal  is  known,  the  area  of 
these  triangles  may  be  found  by  (142). 

This  rule  applies  equally  well  to  a  trapezium. 

Ex.  1.  Find  the  area  of  a  trapezoid  whose  parallel  sides  are 
97  and  84,  and  the  perpendicular  distance  between  them  47 
feet.    :^     U  /  6  ^  ^2-- 

Ex.  2.  Find  the  area  of  a  figure  whose  four  sides  are  succes 
sively  27,  77,  28,  and  85  rods  in  length,  the  angle  between  the 
first  and  second  side  93°,  and  between  the  third  and  fourth 
76°  15'.  Ans.   13  acres,  2  roods,  33.97  sq.  rds. 

Ex.  3.    Find  the  area  of  a  trapezium  whose  sides  are  succes- 
sively 35.8,  13.32,  35.84,  and  17.8  rods,  and  the  line  from  the 
beginning  of  the  first  to  the  end  of  the  second  side  38.9  rods. 
Ans.  3  acres,  1  rood,  36.25-J-  sq.  rds. 

PROBLEM  IV. 

1 46.  To  find  the  area  of  any  polygon. 

If  the  diagonals  necessary  to  divide  the  figure  into  triangles 
have  been  measured,  the  area  can  be  found  by  finding  the  sum 
of  the  areas  of  the  several  triangles. 

When  these  diagonals  are  not  known,  the  method  generally 
used  is  called  the  rectangidar  method.  The  exposition  of  this 
method  belongs  more  properly  under  the  head  of  surveying. 


/ 


64  PLANE  TKIGONOMETRY. 


MISCELLANEOUS  EXAMPLES. 

1.  The  distance  up  the  inclmed  surface  of  a  hill  whose  angle 
of  elevation  is  7°,  is  25  rods ;  the  hill  descends  on  the  other 
side  to  the  same  level,  with  an  inclination  of  15°.  How  many 
pickets  three  inches  wide,  placed  three  inches  apart,  will  it  take 
to  build  a  fence  over  the  hill  1  Ans.  1194. 

2.  Having  measured  a  horizontal  line  from  the  base  of  a  ver- 
tical tower  to  the  distance  of  160  feet,  I  find  that  the  angle  of 
elevation  of  the  top  of  the  tower  is  21°.  What  is  the  height 
of  the  tower  1  Ans.  61.418  ft. 

3.  Having  measured  from  the  base  of  a  hill,  whose  inclina- 
tion is  65°,  25  rods  on  a  horizontal  plane,  I  find  that  the  angle 
of  elevation  of  the  top  is  16°  25^  What  is  the  altitude  of  the 
hill  above  the  horizontal  plane  1  Ans.  8.538  rds. 

4.  Two  observers,  A  and  B,  at  sea,  a  mile  apart,  take  at  the 
same  time  the  angles  of  elevation  of  a  meteor  which  appears 
due  west  of  each.  A  finds  the  angle  21°  50',  B  19°  30'.  '  What 
is  its  altitude  1  Ans.  3.049  ms. 

5.  From  a  steeple  65  feet  above  the  level  of  an  adjacent 
pond,  the  angle  of  depression  of  one  edge  is  40°  10',  of  the 
other  21°  30'.  What  is  the  width  of  the  pond,  and  its  distance 
from  the  church*'?  Ans.  Width  88  ft.,  distance  77  ft. 

6.  When  a  tree  25  feet  high  casts  a  shadow  100  feet  long, 
what  is  the  sun's  altitude  1  Ans.   14°  2'  10". 


7.  From  the  base  of  a  tower,  the  angle  of  elevation  of  the 
vj^  top  of  a  second  tower  is  30°,  and  from  the  top  of  the  first, 
\,            which  is  175  feet  high,  the  angle  of  depression  of  the  top  of 

the  second  is  10°.     If  both  stand  on  the  same  horizontal  plane, 
what  is  the  height  of  the  second]  Ans.   134.05  ft. 

8.  In  the  ruins  of  Persepolis  there  stand  two  upright  col- 


/  /  ^ 

PRACTICAL  APPLICATIONS. 

umns,  one  G4,  the  other  50  feet  flV>r>Arf>  |Ka  -pln^;  m  ^  nm  dq. 
tweeii  these,  on  the  same  plane,  stands  a  statue,  whose  head 
is  86  feet  from  the  summit  of  the  lower,  and  97  feet  from  that 
of  the  higher  column,  and  the  distance  from  the  foot  of  the 
lower  column  to  the  centre  of  the  base  of  the  statue  is  76  feet. 
What  is  the  distance  between  the  tops  of  the  columns  1 

Ans.  157.03  ft.     y^ 

9.  If  the  horizontal  parallax  of  the  sun,  that  is,  the  angle  at 
the  centre  of  the  sun,  subtended  by  the  radius  of  the  earth 
(3962  miles)  is  8".5776  (log.  sine  5.6189407)  what  is  the  dis- 
tance of  the  sun  from  the  earth  1  Ans.  95273760.9  ms. 

10.  If  the  angle  at  the  earth,  subtended  by  the  sun's  diam- 
eter, is  32',  and  the  distance  as  above,  what  is  the  diameter  of 
the  sun]  Ans.  886845.5  ms. 

11.  If  the  moon's  horizontal  parallax  is  57'  9'',  what  is  its 
distance  from  the  earth"?  Ans.  238341.8  ms. 

12.  If  the  moon  subtends  an  angle  at  the  earth  of  31'  7''', 
and  its  distance  is  as  above,  what  is  its  diameter  ] 

Ans.  2157+  nas. 

13.  If  the  annual  parallax,  that  is,  the  angle  at  the  object, 
subtended  by  the  radius  of  the  earth's  orbit,  of  the  nearest 
fixed  star  (a  Centauri)  is  nearly  1"  (log.  sine  4.685575),  what  is 
its  minimum  distance  ]  Ans.   19650000000000. 


147.  The  angle  which  a  line  makes  with 
the  meridian  is  called  the  bearing  of  the 
line.  Thus,  if  N  S  is  the  meridian  and  the 
angle  NAB  40°,  NAC  75°,  SAD  45°, 
the  bearing  is  of 

A  B,  N.  40°  W,  or  of  B  A,  S.   40°  E. 
A  C,  N.  75°  E.,  or  of  C  A,  S.  75°  W. 
AD,  S.  45°  E.,  or  of  D  A,  N.  45°  W. 


66 


PLANE   TRIGONOMETRY. 


When  the  bearing  of  two  lines  which 
form  an  angle  is  given  it  is  easy  to  find  by 
inspection  the  number  of  degrees  in  the 
angle.     Thus,  the  angle 

BAG  =40°  +  75°=  115° 

BAD  =  40°  +  (180  —  45°)  =  175° 

C  A  D  =  180°  —  75°  —  45°  =  60° 


14.  Having  run  a  line  85  rods  S.  45°  E.,  I  came  to  an  impas- 
sable marsh,  and,  seizing  a  man  round  to  the  other  side  of 
the  marsh,  I  placed  him  exactly  in  the  line ;  then,  running 
N.  20°  E.  20  rods,  I  found  that  the  bearing  of  the  man  was 
S.  24°  E. ;  then,  going  to  the  man,  I  ran  the  line  (S.  45°  E.) 
to  the  corner  44  rods  farther.  What  is  the  whole  length  of 
the  line?  Ans.   167.76  rds. 

15.  Wishing  to  find  the  distance  between  two  objects  just 
visible  in  the  distance,  I  measured  a  base  line  100  rods  due  east, 
and,  at  the  west  end  of  the  line,  found  the  bearing  of  both  ob- 
jects ;  one  N.  17°  W.,  the  other  N.  45°  E. ;  at  the  other  end, 
one  N.  23°  30'  W.,  the  other  N.  40°  E.     What  is  the  distance] 

Ans.  871.92  rds. 

16.  Coming  into  a  harbor  I  observed  a  tower,  eastward  of  it 
a  steeple,  and  still  farther  eastward  a  cliff.  The  bearing  of  the 
tower  was  found  to  be  N.  17°  E.,  of  the  steeple  N.  20°  E,  and 
of  the  cliff  N.  26''  30'  E.  From  a  chart,  the  three  objects  were 
found  to  be  in  one  straight  line  ;  from  the  tower  to  the  steeple 
the  distance  was  set  down  as  25  rods,  and  from  the  steeple  to 
the  cliff  54/^  rods.     What  is  my  distance  from  each  object  1 

Ans.  Tower,  477.68  rods. ;  steeple,  477.06  rods. ;  cliff,  480.2 
rods. 


A  TABLE 


OF 


LOGARITHMS  OF  NUMBERS. 


a.7 


N. 

Log. 

i  N. 

Log. 

IN. 

Log. 

|N. 

1    Log. 

1 

0.000000 

26 

1.414973 

51 

1.707570 

76 

1.880814 

2 

0.301030 

27 

1.431364 

52 

1.716003 

77 

1.886491 

3 

0.477121 

28 

1.447158 

53 

1.724276 

78 

1.892095 

4 

0.602060 

29 

1.462398 

54 

1.732394 

79 

1.897627 

5 

0.698970 

30 

1.477121 

55 

1 . 740363 

80 

1.903090 

6 

0.778151 

31 

1.491362 

56 

1.748188 

81 

I  908485 

7 

0.845098 

32 

1.505150 

57 

1.755875 

82 

1.913814 

8 

0.903090 

33 

1.518514 

58 

1.763428 

83 

1.919078 

9 

0.954243 

34 

1.531479 

59 

1.770852 

84 

1.924279 

1ft 

1.000000 

35 

1.544068 

60 

1.778151 

85 

1.929419 

li 

1.041393 

36 

1.556303 

61 

1.785330 

86 

1.934498 

12 

1.079181 

37 

1.568202 

62 

1.792392 

87 

1.939519 

lU 

1.113943 

38 

1.579784 

63 

1.799341 

88 

1.944483 

14 

1.146128 

39 

1.591065 

64 

1.806180. 

89 

1.949390 

15 

1  176091 

40 

1.602060 

65 

1.812913 

90 

1.954243 

16 

1  204120 

41 

1.612784 

66 

1.819544 

91 

1.959041 

17 

1.230449 

42 

1.623249 

67 

1.826075 

92 

1.963788 

18 

1.255273 

43 

1.633468 

68 

1.832509 

93 

1.968483 

19 

1.278754 

44 

1.64.3453 

69 

1.838849 

94 

1.973128 

S<0 

1.301030 

45 

1.653213 

70 

1.845098 

95 

1.977724 

21 

1.322219 

46 

1.662758 

71 

1.851258 

96 

1.982271 

22 

1.342423 

47 

1.672098 

72 

1.857333 

97 

1.986772 

23 

1.361728 

48 

1.681241 

73 

1.863323 

98 

1.991226 

24 

1.3802J1 
1.397540 

49 

1.690196 

74 

1.8692.32 

99 

1.995635 

25 

50 

1.698970 

75 

1.875061 

100 

2.000000 

In  the  following  table,  in  the  last  nine  columns  of  each 
page,  where  the  first  or  leading  figures  change  from  9's  to  O's, 
points  or  dots  are  introduced  instead  of  the  O's  through  the 
rest  of  the  line,  to  catch  the  eye,  and  to  indicate  that  from 
thence  the  annexed  first  two  figures  of  the  Logarithm  in  the 
second  column  stand  in  the  next  lower  line. 


LOGARITHMS   OF  NUMBERS. 


N.  1   0   |l|2|3i4i5i.6i7|8|9|D.  ! 

100 
101 
102 
103 
104 
105 
106 
107 
108 
109 
110 
111 
112 
113 
114 
115 
116 
117 
118 
119 
120 
121 
122 
123 
124 
125 
126 
127 
128 
129 
130 
131 
132 
133 
134 
135 
136 
137 
138 
139 
140 
141 
142 
143 
144 
145 
146 
147 
148 
149 
150 
151 
152 
153 
154 
155 
156 
157 
158 
159 

000000 
4321 
8600 

012837 
7033 

021189 
5306 
9384 

033424 
7426 

0434 
4751 
9026 
3259 
7451 
1603 
.5715 
9789 
3826 
7825 
1787 
.5714 
9606 
3463 
7286 
1075 
4832 
8557 
2250 
5912 

0868 
5181 
9451 
3680 
7868 
2016 
6125 
.195 
4227 
8223 
2182 
6105 
9993 
3846 
7666 
1452 
.5206 
8928 
2617 
6276 
9904 
3503 
7071 
.611 
4122 
7604 
1059 
4487 
7888 
1263 
4611 
7934 
1231 
4504 
7753 
0977 
4177 
7354 
.508 
3639 
6748 
9835 
2900 
5943 
8965 
1967 
4947 
7908 
0848 
3769 
6670 
9552 
2415 
5259 
80S4 
0892 
3681 
6453 
9206 
1943 

1301 
5609 
9876 
1100 

8284 
2428 
6533 
.600 
4628 
8620 

2576 
6495 
.380 
4230 
8046 
1829 
5580 
9298 
2985 
6640 
.266 
3861 
7426 
.963 
4471 
7951 
1403 
4828 
8227 
1599 
4944 
8265 
1560 
4830 
8076 
1298 
4496 
7671 
.822 
3951 
7058 
.142 
3205 
6246 
9266 
2266 
5244 
8203 
1141 
4060 
6959 
9839 
2700 
5542 
8366 
1171 
3959 
6729 
9481 
2216 

1734 
6038 
.300 
4521 
8700 
2841 
6942 
1004 
5029 
9017 
2969 
6885 
.766 
4613 
8426 
2206 
5953 
9668 
3352 
7004 
.626 
4219 
7781 
1315 
4820 
8298 
1747 
5169 
8565 
1934 
5278 
8595 
1888 
5156 
8399 
1619 
4814 
7987 
1136 
4263 
7367 
.449 
3510 
6549 
9567 
2564 
5541 
8497 
1434 
4351 
7248 
.126 
2985 
5825 
8647 
1451 
4237 
7005 
9755 
2488 

2166 
6466 
.724 
4910 
9116 
3252 
7350 
1408 
5430 
9414 
3362 
7275 
11.53 
4996 
8805 
2582 
6326 
..38 
3718 
7368 
.987 
4576 
8136 
1667 
5169 
8644 
2091 
5510 
8903 
2270 
.5611 
8926 
2216 
.5481 
8722 
1939 
5133 
8303 
1450 
4574 
7676 
.756 
3815 
6852 
9868 
2863 
5838 
8792 
1726 
4641 
7536 
.413 
3270 
6108 
8928 
1730 
4514 
7281 
..29 
2761 

2598 
6894 
1147 
5360 
9532 
3664 
7757 
1812 
5830 
9811 
3755 
7664 
1538 
5378 
9185 
2958 
6699 
.407 
4085 
7731 
1347 
4934 
8490 
2018 
.5518 
8990 
2434 
5851 
9241 
2605 

5943 
9256 
2544 
5806 
9045 
2260 
.5451 
8618 
1763 
4885 
7985 
1063 
4120 
7154 
.168 
3161 
6134 
9086 
2019 
4932 
7825 
.699 
3555 
6391 
9209 
2010 
4792 
7556 
.303 
3033 

3029 
7321 
1570 
5779 
9947 
4075 
8164 
2216 
6230 
.207 

3461 
7748 
1993 
6197 
.361 
4486 
8571 
2619 
6629 
.602 
4540 
8442 
2309 
6142 
9942 
3709 
7443 
1145 
4816 
8457 
2067 
5647 
9198 
2721 
6215 
9681 
3119 
6531 
9916 
3275 
660i 
9915 
3198 
6456 
9690 
2900 
6086 
9249 
2389 
5507 
8603 
1676 
4728 
7759 
.769 
3758 
6726 
9674 
2603 
.5512 
8401 
1272 
4123 
6956 
9771 
2567 
5346 
8107 
.8.50 
3577 

3891 
8174 
2415 
6616 
.775 
4896 
8978 
3021 
7028 
.998 

4932 
8830 
2694 
6524 
.320 
4083 
7815 
1514 
5182 
8819 
2426 
6004 
95.52 
3071 
6562 
..26 
3462 
6871 
.253 
3600 
6940 
.245 
3525 
6781 
..12 
3219 
6403 
9564 
2702 
.5818 
8911 
1982 
.5032 
8061 
1068 
4055 
7022 
9968 
2895 
.5802 
8689 
1.5.58 
4107 
'239 
..51 
2846 
5623 
8382 
1124 
3848 

432 
428 
424 
419 
416 
412 
408 
404 
400 
396 
393 
389 
386 
382 
379 
376 
372 
369 
366 
363 
360 
357 
355 
351 
349 
346 
343 
340 
338 
335 
333 
330 
328 
325 
323 
321 
318 
315 
314 
311 
309 
307 
305 
303 
301 
299 
297 
295 
293 
291 
289 
287 
285 
283 
281 
279 
278 
276 
274 
272 

041393 
5323 
9218 

053078 
6905 

000698 
4458 
8186 

071882 
5547 

4148 
8053 
1924 
5760 
9563 
3333 
7071 
.776 
4451 
8094 
1707 
5291 
8845 
2370 
.5866 
9335 
2777 
6191 
9579 
2940 

6276 
9586 
2871 
6131 
9368 
2580 
5709 
8934 
2076 
5196 
8294 
1370 
4424 
7457 
.469 
3460 
6430 
9380 
2311 
5222 
8113 
.985 
3839 
6674 
9490 
2289 
5069 
7832 
.577 
3305 

079181 

082785 
6360 
9905 

093422 
6910 

100371 
3804 
7210 

110590 

9543 
3144 
6716 
.258 
3772 
7257 
0715 
4140 
7549 
0926 
4277 
7603 
0903 
4178 
7429 
06,55 
3858 
7037 
.194 
3327 
6438 
9527 
2594 
5640 
8664 
1667 
4650 
7613 
0555 
3478 
6381 
9264 
2129 
4975 
7803 
0612 
3403 
6176 
8932 
1670 

11.3943 
7271 

120574 
3852 
7105 

130334 
3539 
6721 
9879 

143015 

146128 
9219 

152288 
5336 
8362 

161368 
4353 
7317 

170262 
3186 

176091 
8977 

181844 
4691 
7.521 

190332 
3125 
.5899 
8657 

201397 

N.  1   0   1  1  1  2  1  3  1  4  1  5  1  6  1  7  1  8  1  9  1  D. 

2^ 


LOGARITHMS  OF  NUMBERS. 

3 

Y  jV. 

^        A       /I      y^         >  

— ~.~_.^ 

[nTI 

0   1  1  |cilV7  4fl|^  176  Pl    \    S    \    9ND.1 

160 

204120  4391 

>1*63 

493^1  52iR 

5475  5746 

6016 

6286, 6556 

271 

161 

6826 

7096 

7365 

7634 

7904 

8173  8441 

8710 

8970 

9247 

269 

162 

9515 

9783 

..51 

.319 

.586 

.853  1121 

1388 

1654 

1921 

267 

163 

212188 

2454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

266 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

165 

7484 

7747 

8010 

8273 

8536 

8798 

9060 

9323 

9585 

9846 

262 

166 

220108 

0370 

0631 

0892 

1153 

1414 

1675 

1936 

2196 

2456 

261 

167 

2716 

2976 

3236 

3496 

3755 

4015 

4g74 

4533 

4792 

5051 

259 

168 

6309 

5568 

5826 

6084 

6342 

6600 

6858 

7115 

7372 

7630 

258 

169 
170 

7887 

8144 
0704 

8400 
0960 

8657 
1215 

8913 
1470 

9170 
1724 

9426 
1979 

9682 
2234 

9938 

2488 

.193 

2742 

256 
254 

23U449 

171 

2996 

3250 

3504 

3757 

4011 

4264 

4517 

4770 

5023 

5276 

253 

172 

5528 

5781 

6033 

6285 

6537 

6789 

7041 

7292 

7544 

7/95 

252 

173 

8046 

8297 

8548 

8799 

9049 

9299 

9550 

9800 

...50 

.300 

250 

174 

240549 

0799 

1048 

1297 

1546 

1795 

2044 

2293 

2541 

2790 

249' 

175 

3038 

3286 

3534 

3782 

4030 

4277 

4525 

4772 

.5019 

5266 

2433 

176 

5513 

5759 

6006 

6252 

6499 

6745 

6991 

7237 

7482 

7728 

246 

177 

7973 

8219 

8464 

8709 

8954 

9198 

9443 

9687 

9932 

.176 

245 

178 

250420 

0664 

0908 

ri5i 

1395 

1638 

1881 

2125 

2368 

2610 

243 

179 

180 

2853 

3096 
5514 

3338 
5755 

3580 
5996 

3822 
6237 

4064 
6477 

4306 
6718 

4548 
6958 

4790 
7198 

5031 

7439 

242 
241 

255273 

181 

7679 

7918 

8158 

8398 

8637 

8877 

9116 

9355 

9594 

9833 

239 

182 

260071 

0310 

0548 

0787 

1025 

1263 

1501 

1739 

1976 

2214 

238 

183 

2451 

2688 

2925 

3162 

3399 

3636 

3873 

4109 

4346 

4582 

237 

184 

4818 

5054 

5290 

5525 

5761 

5996 

6232 

6467 

6702 

6937 

235 

185 

7172 

7406 

7641 

•;'875 

8110 

8344 

8578 

8812 

9046 

9279 

234 

186 

9513 

9746 

9980 

.213 

.446 

.679 

.912 

1144 

1377 

1609 

233 

187 

271842 

2074 

2306 

2538 

2770 

3001 

3233 

34M 
5772 

3696 

3927 

232 

188 

4158 

4389 

4620 

4850 

5081 

5311 

5542 

6002 

■6232 

230 

189 
190 

6462 

6692 

6921 
9211 

7151 
9439 

7380 
9667 

7609 
9895 

7838 
.  123 

8067 
.351 

8296 
.578 

8525 

.806 

229 

228 

278754 

8982 

191 

281033 

1261 

1488 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 

192 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

226 

193 

5557 

5782 

6007 

6232 

6456 

6681 

6905 

7130 

7354 

7578 

225 

194 

7802 

802b 

8249 

8473 

8696 

8920 

9143 

9366 

9589 

9812 

223 

195 

200035 

0257 

0480 

0702 

0925 

1147 

1369 

1591 

1813 

2034 

222 

196 

2256 

2478 

2699 

2920 

3141 

3363 

3584 

3804 

4025 

4246 

221 

197 

4466 

4687 

4907 

5127 

5347 

5567 

5787 

6007 

6226 

6446 

220 

198 

6665  6884 

7104 

7323 

7542 

7761 

7979 

8198 

8416 

8635 

219 

199 

8853 

9071 

9289 

9507 

9725 

9943 

.161 

.378 

.595 

.813 

218 

200 

30lOoO 

1247 

1464 

1681 

1898 

2114 

2331 

2547 

2764 

2980 

217 

201 

3196 

3412 

3628 

3844 

4059 

4275 

4491 

4706 

4921 

5136 

216 

202 

5351 

5566 

5781 

5996 

6211 

6425 

6639 

6854 

7068 

7282 

215 

203 

7496 

7710 

7924 

8137 

8.351 

8564 

8778 

8991 

9204 

9417 

213 

204 

9630 

9843 

..56 

.268 

.481 

.693 

.906 

1118 

1330 

1.542 

212 

205 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

211 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

210 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

7646 

7854 

209 

208 

8063 

8272 

8481 

8689 

8898 

9106 

9314 

9522 

9730 

9938 

208 

209 
210" 

320146 

0354 
2426 

0562 
2633 

0769 
2839 

0977 
3046 

1184 
3252 

1391 
3458 

1598 
3665 

1805 
3871 

2012 
4077 

207 
206 

322219 

211 

4282 

4488 

4694 

4899 

5105 

5310 

5516 

5721 

5926 

6131 

205 

213 

6336 

6541 

6745 

6950 

7155 

7359 

7563 

7767 

7972 

8176 

204 

213 

8380 

8583 

8787 

8991 

9194 

9398 

9601 

9805 

...8 

.211 

203 

214 

330414 

0617 

0819 

1022 

1225 

1427 

1630 

1832 

2034 

2236 

202 

215 

2438 

2640 

2842 

3044 

3246 

3447 

3649 

3850 

4051 

4253 

202 

216 

4454 

4655 

4856 

5057 

5257 

5458 

5658 

5859 

6059 

6260 

201 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

7858 

8058 

8257 

200 

218 

84561  8656 

8855 

9054 

9253 

9451 

9650 

9349 

..47 

.246 

199 

219 

340W4I  0642 

0841 

1039'  1237 

1435 

1632'  1830' 2028' 2225'  198  j 

!  N.  1   0   1  1  1  2  '  3  i  4  1  5  1  6  !i  7  1  8  i  9  1  n.  1 

'tip 


LOGARITHMS   OF  NUMBERS. 


' 

N. 

1   0   1  1  [  2  1  .3  1  4  1  5  1  6  1  7  1  8  1  9  1  D. 

^20" 

342423.  2620|  2817| 3014  3212, 3409, 3606  3802, 3999i  4l96i  197 

\f     A  . 

221 

4392  4589)4785 

4981 

5178  5374 

6570  5766 

5962 

6157 

196 

^  \  ^ 

222 

6353  6549  6744 

6939 

7135  7330 

7526 i  7720 

7915 

8110 

195 

L  T  r 

223 

8305  8500|  8694 

8889 

9083, 9278 

9472, 9666 

9860 

...54 

194 

IS5  \ 

224 

350248  0442 

0636 

0829 

1023  1216 

1410  1603 

1796 

1989 

193 

f  K  V 

225 

2183;  2375 

2568 

2761 

2954  3147 

3339^3532 

3724 

3916 

193 

^< 

226 

4i08|  4301 

4493 

4685'  4876 

6068 

6260, 6452 

5643 

6834 

192 

227 

6026!  6217 

6408 

6599,  6790 

6981 

7172  7363 

7654 

7744 

191 

^28 

7935!  8125 

8316 

8506 

8696 

8886 

9076  9266 

9456 

9646 

190 

i 

229 
23C 

9835 

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1917 

.215 
2105 

.404 
2294 

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2482 

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2671 

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2859  i  .3048 

1350 
3236 

1.539!  189 

3424 1  188 

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361728 

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3612 

3800 

3988 

4176 

4363 

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6049 

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6423 

6610 

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J-  \  ^ 

233 

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234 

9216 

9401 

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9772 

9958 

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185 

235 

371068!  1253 

1437 

1622 

1806 

1991 

2176 

2360 

2544 

2728 

184 

236 

2912 

30U6 

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3464 

3047 

3831 

4015 

4198 

4382 

4565! 1841 

V  »  jo 

237 

4748 

4932 

5115 

5298 

5481 

5664 

5846 

6029 

6212 

6394 

183 

i  \ 

238 

6577 

6759 

6942 

7124 

7306 

7488 

7670!  78.52 

8034 

8216 

182 

239 
240 

8398 

8580 
039;i 

8761 

8943 
0754 

9124 
0934 

9306 
1115 

9487  9668 
1296  1476 

9849 
1656 

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1837 

181 
181 

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0573 

1  \\   V 

241 

2017 

2197 

2377 

2557 

2737 

2917 

3097 

3277 

3456 

3636 

180 

V  \\  ^ 

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4174 

4353 

4533 

4712 

4891 

6070 

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5428 

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i  V  ' 

243 

5606 

5785 

5964 

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6499 

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178 

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244 

7390 

7568 

7746 

7923 

8101 

8279 

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8634 

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245 

9166 

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9520 

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9875 

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246 

390935 

1112 

1288 

1464 

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1993 

2169 

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247 

2697 

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3048 

3224 

3400 

3575 

3751 

3926 

4101 

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248 

4452 

4627 

4802 

4977 

5152 

5326 

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249 
250 

6199 

6374 

6548 

8287 

6722 
8461 

6896 
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7071 
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7245 

8981 

7419 
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7592 
9328 

7766 
9501 

174 
173 

397940 

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251 

9674 

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.192 

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1066 

12281  173 1 

252 

401401 

1573 

1745 

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2089 

2261 

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26051 2777 

2949 

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253 

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254 

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257 

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258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

168 

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259 
260 

3300 

3467 
5140 

3635 
5307 

3803 
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3970 
5641 

4.37 
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5974 

4472 
6141 

4639 
6308 

4806 
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167 
167 

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414973 

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261 

6641 

6807 

6973 

7139 

7306 

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7638 

7804 

7970 

8135  166 

i  ^  A 

262 

8301 

8467 

8633 

8798 

8964 

9129 

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9460 

9625 

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V  t   V 

263 

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264 

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1933 

2097 

2261 

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265 

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3737 

3901 

4065 

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In,     V 

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266 

4882 

5045 

5208 

5371 

5534 

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5860 

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267 

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4     iv  i 

268 

8135 

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8783 

8944 

9106 

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t   vVi 

269 

9752  9914 

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1042 

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1685 

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3450 

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3770 

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HV 

272 

4569!  4729' 4888!  50481 

6207 

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5526!  6685'  6844 

6004 

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6163  6322i 6481  6640 

6798! 6957 

7116172751  7433  7592 

159 

274 

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8384  8542 

870118859  9017  9175 

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N.  I  1)   I  I  I  2  i  3 


280 

447158 

7313 

281 

8706 

8861 

282 

450249 

0403 

283 

1786 

1940 

284 

3318 

3471 

285 

4845 

4997 

286 

6366 

6518 

287 

7882 

8033 

288 

9392 

9543 

289 

460898 

1048 

290 

462398  25481 

291 

3893 

4042' 

292 

5383 

5532 

293 

6868 

7016 

294 

8347 

8495 

295 

9822 

9969 

296 
297 

471292 
27)56 

1^38 
2903 

298 

4216 

4362 

299 

5671 

5816 
7266 

300 

477121 

301 

8566 

8711 

302 

480007 

0151 

303 

1443 

1586 

304 

2874 

3016 

305 

4300 

4442 

306 

5721 

5863 

307 

7138 

7280 

308 

8551 

8692 

309 

9958 

..99 

310 

491362 

1502 

311 

2760 

2900 

312 

4155 

4294 

313 

5544 

5683 

314 

6930 

7068 

315 

8311 

8448 

316 

9687 

9824 

317 

501059 

1196 

318 

2427 

2564 

319 

3791 

3927 

320 

505150 

5286 

321 

6505 

6640 

322 

7856 

7991 

323 

9203 

9337 

324 

510545 

0679 

325 

1883 

2017 

326 

1218 

3351 

327 

4548 

4681 

323 

5874 

6006 

329 

7196 

7328 

330 

618514 

8646 

331 

9828 

99.59 

332 

521138 

1269 

333 

2444 

2575 

334 

3746 

3876 

335 

6045 

5I74| 

336 

633Q 

6469 

33/ 

7630 

7759 

338 

8917 

9045 

339 

530200 

0328' 

7468 
9015 
0557 
2093 
3624 
5150 
6670 
8184 
9694 
1198 
2697 
4191 
5680 
7164 
8643 
.116 
1585 
3049 
4508 
6962 


7411 
8855 
0294 
1729 
3159 
4585 
6005 
7421 
8833 
.239 


1642 
3040 
4433 
5822 
7206 
8586 
9962 
1333 
2700 
4063 
5421 
6776 
8126 
9471 
0813 
2151 
3484 
4813 
6139 
7460 


8777 

..90 

1400 

270 

4006 

5304 

6598 

7888 

9174 

0456 


LiLi 


I  1 


.5557 

691 

8260 

9606 

0947 

2284 

3617 

4946 

6271 

7592! 

8909 

.221 

1530 

2835 

4136 

5434 

6727 

8016 

9302 

0584 


LOGARITHMS   OF  NUMBERS. 


~N. 

1   0   |l|2|3|4|5|6|7|8!9|D.  ! 

340' 

531479 

1607 

i:34 

1862 

1990 

211712245 

2372 

2500 

2627 

128 

341 

2754 

2882 

3009 

3136 

3264 

3391 

3518 

3645 

3772 

3899 

127 

342 

4026 

4153 

4280 

4407 

4534 

4661 

4787 

4914 

5041 

5167 

127 

343 

5294 

5421 

5547 

5674 

5800 

5927 

6053 

6180 

6306 

6432 

126 

344 

6558 

6685 

6811 

6937 

7063 

7189 

7315 

7441 

7567 

7693 

126 

345 

7819 

7945 

8071 

8197 

8322 

8448 

8574 

8699 

8825 

8951 

126 

346 

9076 

9202 

9327 

9452 

9578 

9703 

9829 

9954 

..79 

.204 

125 

347 

540329 

0455 

0580 

0705 

0830 

0955 

1080 

1205 

1330 

1454 

123 

348 

1579 

1704 

1829 

1953 

2078 

2203 

2327 

2452 

2576 

2701 

125 

349 
350 

2825 

2950 
4192 

3074 
4316 

3199 
4440 

3323 

3447 

4688 

3571 

3696 
4936 

3820 
5060 

3944 
5183 

124 
124 

544068 

4564 

4812 

351 

5307 

5431 

5555 

5678 

5802 

5925 

6049 

6172 

6296 

6419 

124 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

123 

353 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

123 

354 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

.106 

123 

355 

550228 

0351 

0473 

0595 

0717 

0840 

0902 

1084 

1206 

1328 

122 

356 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

122 

357 

2068 

2790 

2911 

3033 

3155 

3276 

3398 

3519 

3640 

3762 

121 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

121 

359 

360 

5094 

5215 

5336 

5457 
6664 

5578 

6785 

5699 
6905 

5820 
7026 

5940 

6061 
7267 

6182 
7387 

121 
120 

550303 

6423 

6544 

7146 

361 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

120 

362 

8709 

8829 

8948 

9068 

9188 

9308 

9428 

9548 

9667 

9787 

120 

363 

9907 

..26 

.146 

.265 

.385 

.504 

.624 

.743 

.863 

.982 

119 

364 

561101 

1221 

1340 

1459 

1578 

1698 

1817 

1936 

2055 

2174 

119 

365 

2293 

2412 

2531 

2650 

2769 

2887 

3006 

3125 

3244 

3362 

119 

366 

5481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

119 

367 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

US 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

6673 

6791 

6909 

118 

369 
370 

7026 
568202 

7144 
8319 

7262 
8436 

7379 

8554 

7497 
8671 

7614 

8788 

7732 

8905 

7849 

7967 

8084 
9257 

118 
117 

9023 

9140 

371 

9374 

9491 

9608 

9725 

9842 

9959 

..76 

.193  .309 

.426 

117 

372 

570543 

0660 

0776 

0893 

1010 

1126 

1243 

1359 

1476 

1592 

117 

373 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2523 

2639 

2755 

M6 

374 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3684 

3800 

3915 

116 

375 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

116 

376 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

115 

377 

6341 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

115 

378 

7492 

7607 

7722 

7836 

7951 

8006 

8181 

8295 

8410 

8525 

115) 

379 
380 

8639 
579784 

8754 
9898 

8868 
..12 

8983 
.126 

9097 
.241 

9212 
.355 

9326 
.469 

9441 

9555 
.697 

9609 
.811 

114 
114 

.583 

381 

580925 

1039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

114 

382 

2063 

2177 

2291 

2404 

2518 

2631 

2745 

2858 

2972 

3085 

114 

383 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

113 

384 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

5122 

5235 

5348 

113 

385 

6461 

5574 

5686 

5799 

5912 

6024 

6137 

6250 

6362 

6475 

113 

386 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

112 

387 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

S608 

8720 

112 

388 

8832 

8944 

9056 

9167 

9279 

9391 

9503 

9615 

9726 

9838 

112 

389 
390 

9950 

..61 

.173 

1287 

.284 
1399 

.396 
1510 

.507 

.619 
1732 

.730 

.842 
1955 

.953 
2066 

112 
111 

691065 

1176 

1021 

1843 

391 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

111 

392 

3286 

3397 

3508 

3618 

3729 

3S40 

3950 

4061 

4171 

4282 

111 

393 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

110 

394 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

626V 

6377 

6487 

110 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

110 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

110 

397 

8791 

8900 

9009 

9119 

9228  9337 

9446 

9556 

9665  9774 

109 

398 

9883 

9992 

.101 

.210  .3191 .428 

.537 

.646 

.755  .864 

109 

399 

600973 

10S2 

1191 

1299  14081  1517 

1625 

1734 

1843  1951 

109 

N.  1 

0   Il|2|3i4l5i6|7l8|9in.  J 

LOGARITHMS   OF  NUMBERS. 


N.~ 

1   0   1  1  1  2  1  3  !  4  1  .  5  1  6  1  7  !  8  1  9  I  D  ! 

400 

602060 

1  2169 

2277 

2386,2494,26031  2711 

2819 

2928 

3036'  JOS' 

401 

3144 

J253 

3361 

3469 

3.577 

3686 

3794 

3902 

4010 

4118 

108 

102 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

6089 

5197 

108 

403 

5305 

5413 

5521 

5628 

5736 

5844 

.5951 

6059 

6166 

6274 

108 

404 

635r 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7.3481  107| 

405 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8313 

8419 

107 

406 

8526 

8633 

8740 

8847 

8954 1 9061 

9167 

9274 

9381 

9488 

107 

407 

95941  9701 

9808 

9914 

..2li  .128 

.234 

.341 

.447 

..5.54 

107 

408 

610660 
1723 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

106 

409 
410 

1829 
2890 

1936 
2996 

2042 
3102 

2148 
3207 

22h4 
3311 

2360 
.3419 

2466 
3525 

2572 
36.30 

2678 

106 

612784 

3736  106 

4'.: 

3842 

3947 

4053 

41.59 

4264 

4370 

4475 

4.581 

4686 

4792  106 

412 

4897 

5003 

5108 

5213 

5319 

5424 

5529 

5634 

5740 

.58451  105 

413 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6790 

68951  105 

414 

7000 

7105 

7210 

7315 

7420 

7525 

7629 

7734 

7839 

79431  105 

415 

8048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

105 

116 

9093 

9198 

9302 

9406 

951  ll  9615 

9719 

9824 

9928 

..32 

104 

417 

620136 

0240 

0344 

0448 

0552 

0656 

0760 

0864 

0968 

1072 

101 

418 

1176 

1280 

1384 

1488 

1592 

1695 

1799  1903 

2007 

2110 

104 

419 
420 

2214 

2318 
3353 

2421 
3456 

2525 
3559 

2628 
3663 

2732 
3766 

2835  2939 

3042 

3146 

104 

623249 

3809 

3973 

407614179!  103 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

5004 

61071.52101  103 

422 

5312 

5415 

5518 

5021 

5724 

58271  5929 

6032 

6 1 35 1 6338  103 

423 

6340 

6443 

6546 

6648:6751 

6853!  6956 

7058 

7161  72631  103 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8185  82871  102 

425 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206  93081  102 

426 

9410 

9512 

9613 

9715 

9817 

9919 

..21 

.123 

.2241 .326 

102 

427 

630428 

0530 

0631 

0733 

0835 

0936 

1038 

1139 

12411  1342 

102 

428 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

21.53 

2255! 2356 

101 

429 

2457 

2559 

2660 

2761 

2862 

2963 

.3064 

3105 

3266 i  3367 

101 

430 

633468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

100 

431 

4-177 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

5383  100 

432 

5484 

5584 

5685 

5785 

.5886 

5986 

6087 

6187 

6287 

6388  100 

43'3 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290] 7390 

100 

434 

7490 

7590 

7690 

7790 

7890 

7990 

8090 

8190 

82901 8389 

99 

435 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287^  9387 

99 

436 

9486 

9586 

9686 

9785 

9885 

9984 

..84 

.183 

.283 

.382 

99 

437 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

99 

438 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

99 

439 

440 

2465 

2563 
3551 

2662 
3650 

2761 
3749 

2860 
3847 

2959 
3946 

3058 
4044 

3156 
4143 

3255  3354 
4242  4340 

99 
98 

643453 

441 

^  4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

5226  5324 

98 

442 

5422 

5521 

6619 

5717 

5815 

.5913 

6011 

6110 

6208 

6306 

93 

443 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7089 

7187 

7285 

98 

444 

7383  7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

98 

445 

smo 

8458 

85.55 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

97 

lie 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

..16 

.113 

.210 

97 

117 

650308 

0405 

0502 

0.599 

0696 

0793 

0890 

0987 

1084 

1181 

97 

HS 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

19.56 

2053 

2150 

97 

149 

2246 

2343 

2440 

2536 

2633 

2730 

2826 

2923 

3019|3I16 

97 

450 

C5?i\3 

3309 

3405 

3502 

.35981  3695 

3791 

3888 

3984! 4080 

96 

451 

4H7 

4273 

4369 

4465 

4562 

4658 

4754 

4850 

494 6  .50  42 

96 

452 

5138 

5235 

.5331 

5427 

.5523 

5619 

5715 

.5810 

.5906  60021  961 

453 

5098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

69601  96  1 

454 

7056 

7152' 

7247 

7343 

74381  7534 

7629  7725 

7820 

7916 

96 

455 

8011 

8107 

8202 

8298 

8393 

8488 

8.584 

6679 

8774 

8870 

95 

456 

8965 

9060 

9155 

9250 

9346 

944 1 

9536 

9631 

9726 

9821 

95 

457 

9916 

..11 

.106 

.201 

.296 

.391 

.486 

..581 

.676 

.771 

95 

458 

660895 

0960 

1055 

11.50 

1245 

1.339 

14,34 

1529 

1623 

1718 

95 

459 

1813 

1907  2002 

2096 

2191 

228C 

2380 

2475 

2569 

2663  95 1 

^ 

0   1  1  1  2  i  3  1  4  1  5  1  6  1  7  i  8  1  9  1  D.  i 

U  0 


LOGARITHMS   OF  NUMBERS. 


-^\ 

0   ;i|2|3|4!5|6|7|8|9!D.  1 

460 

662758 

2852| 2947 1  3041, 

3135  3230,  3324,  34l8i  3512|  3607 

94 

461 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

94 

462 

4642 

4736 

4830 

4924 

5018 

5112 

5200 

5299 

5393 

.6487 

94 

463 

»581 

5675 

5769 

5862 

5956 

6050 

6143 

6237 

6331 

6424 

94 

464 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

94 

165 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

93 

466 

8386 

8479 

9572 

8665 

8759 

8852 

8945 

9038 

9131 

9224 

93 

467 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

..60 

.153 

93 

468 

670246 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

1080 

93 

469 
170 

1173 

1265 
2190 

1358 
2283 

1451 

1543 

1636 
2560 

1728 
2652 

1821 
2744 

1913 

2005 

93 
92 

072098 

2375 

2467 

2836 

2929 

471 
472 

b02l 
3942 

3113 
4034 

3205 
4126 

3297 
4218 

3390 
4310 

3482 
4402 

3574 

3666 
4586 

37581  3850 
4677,  4769 

92 
92 

Vm 

473 

486 1 

4953 

6045 

5137 

5228 

5320 

5412 

5503 

5595 

5687 

92 

174 

5778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

92 

175 

6694 

6785 

6870 

6968 

7059 

7151  7242 

7333 

7424 

7616 

91 

476 

7607 

7698 

7789 

7881 

7972 

8063 

8154 

8246 

8336 

8427 

91 

477 

8518 

8609 

8700 

8791 

8882 

8973 

9064 

9165 

9246 

9337 

91 

178 

9428 

9519 

9610 

9700 

9791 

9882 

9973? 

..63 

.1.64 

.245 

91 

479 

680336 

0426 

0517 

0607 

0698 

0789 

0879 

0970 

1060 

1151 

91 

480 

681241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2066 

90 

481 

2145 

2235 

2326 

2416 

2506 

2596 

2686 

2777 

2867 

2957 

90 

482 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

90 

483 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4766 

90 

484 

4845 

4935 

5025 

5114 

5204 

5294 

5383 

5473 

5563 

5662 

90 

485 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

89 

486 

6036 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

89 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

89 

488 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

89 

489 
490 

9309 
690196 

9398 
0285 

9486 
0373 

9575 

9664 
0550 

9753 
0639 

9841 

9930 

..19 

.107 

89 
89 

0462 

0728 

0816 

0905 

0993 

491 

1081 

1170 

1258 

1347 

1435 

1624 

1612 

1700 

1789 

1877 

88 

492 

1965 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2769 

98 

493 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3651 

3639 

98 

494 

3727 

3815 

3903 

3991 

4078 

4166 

4254 

4342 

4430 

4517 

88 

495 

4605 

4093 

4781 

4868 

4956 

5044 

5131 

5219 

5307 

5394 

88 

496 

5482 

5569 

5657 

5744 

5832 

5919 

6007 

6094 

6182 

6269 

87 

497 

6356 

6444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

87 

498 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

87 

499 
500 

8101 
698970 

8188 

8275 
9144 

8362 
9231 

8449 
9317 

8535 
9404 

8622 
9491 

8709 
9578 

8796 
9664 

8883 
9751 

87 
87 

9057 

501 

9838 

9924 

..11 

..98 

.184 

.271 

.368 

.444 

.531 

.617 

87 

502 

700704 

0790 

0877 

0963 

1050 

1130 

1222 

1309 

1395 

1482 

86 

503 

1568 

1654 

1741 

1827 

1913 

1999 

2086 

2172 

2258 

2344 

86 

504 

2431 

2517 

2603 

2689 

2775 

2861 

2947 

3033 

3119 

3205 

86 

50£ 

3291 

3377 

3463 

3549 

3635 

3721 

3807 

3895 

3979 

4066 

86 

596 

4151 

4236 

4322 

4408 

4494 

4579 

4665 

4751 

4837 

4922 

86 

507 

6008 

5Qtf4 

5179 

5265 

5350 

5436 

5522 

5607 

5693 

6778 

86 

503 

5864 

5949 

C035 

6120 

6206 

6291 

6376 

6462 

6547 

6632 

85 

509 
510 

6718 
707570 

6803 
7656 

6888 

6974 

7826 

7059 

7144 
7996 

7229 
8081 

7316 
8166 

7400 
8251 

7485 

86 

85 

7740 

7911 

8336 

511 

8421 

8506 

8591 

8676 

8761 

8846 

8931 

9015 
9863 

9100 

9185 

85 

512 

9270 

9355 

9440 

9524 

9609 

9694 

9779 

9948 

..33 

85 

513 

710117 

0202 

0287 

0371 

0456 

0540 

0625 

0710 

0794 

0879 

86 

514 

0963 

1048 

1132 

1217 

1301 

1385 

1470 

1.554 

1639 

1723 

84 

515 

1807 

1892 

1976 

2060 

2144 

2229 

2313 

2397 

2481 

26C6 

84 

516 

2650 

2734 

2818 

2902 

2986 

30701 3154 

3238 

3323 

3407 

84 

517 

3491 

3575 

3650 

3742 

3826 

3910  3994 

4078 

4162 

4246 

84 

518 

4330 

4414 

4497 

4581 

4665 

4749  4833 

4916 

5000 

5084 

84 

519 

5167 

5251 

53351  5418 

5502 

5586  5669 

5753'  5836 

5920 

841 

'nT 

1   0   1  1  1  2  1  3  1  4  1  5  1  6  1  7  1  8  1  9  1  D.l 

LOGARITHMS   OF  NUMBERS. 


'^nT 

1   0   {1|2|3|4|5|6|7|8|9|D^ 

520 

716003 

6087  6170 

62.54 

6337 

6421 

6.504,  65«8|  6671 

67.54 

^3 

521 

6838 

6921  7004 

7088 

7171 

72.54 

73381 7421  7504 

7587 

83 

522 

7671 

7754  7837 

7920 

8003 

8086 

8169| 8253; 8336 

8419 

83 

523 

8502 

8585  8668 

8751 

8834 

;8917 

9000, 9083!  9165 

9218 

83 

524 

9331 

9414  9497 

9580 

9663 

i  9745 

982819911 

9994 

..77 

83 

525 

720159 

0242, 0325 

0407 

0490 

;  0573i  0655|  0738 

0821 

0903 

83 

526 

0986 

1068) 1151 

12.33 

1310 

1398 

1481  1.563 

1646 

1728 

82 

527 

1811 

1893  1975 

2058 

2140 

2222 

2305i  2387 

2469 

2552 

82 

528 

2634 

2716! 2798 

2881 

2963 

3045 

3127 

3209 

3291 

,337 1 

82 

529 

3456 

3538, 3620 

3702 

3784 

3866 

3948 

4030 

4112 

4194 

82 

530 

724276 

4358' 4440 

4522 

4604 

4685 

4767 

4849 

4931 

.5013 

82 

531 

5095 

5 1 76 1  .5258 

5340 

5422 

5503 

5585 

5667 

6748 

6830 

82 

532 

5912 

59931  6075 

61.56 

6238 

6320 

6401 

6483 

6564 

6646 

82 

533 

6727 

6809;  6890 

6972 

7053 

7134 

7216 

7297 

7379 

7460 

81 

534 

7541 

76231 7704 

7785 

7866 

7948 

8029 

8110 

8191 

8273 

81 

535 

8354  843518516 

8597 

8678 

8759 

8841 

8922 

9003 

9084 

81 

536 

9165  9246,9327 

9408 

9489 

9570 

9651 

9732 

9813 

9893 

81 

537 

9974  ..55  .136 

.217 

.298 

.378 

.459 

•540 

.621 

.702 

81 

538 

730782  0863  0944 

1024 

1105 

1186 

1266 

1347 

1428 

1508 

81 

539 
540 

1589  1C69  1750 

1830 
2635 

1911 

1991 
2796 

2072 

2152 
2956 

2233 
3037 

2313 
3117 

81 
'80 

732394 

2474 

2555 

2715 

2876 

541 

3197 

3278 

3358 

34.38 

3518 

3598'  3679 

3759 

3839 

3919 

80 

542 

3999 

4079 

4160 

4240 

4320 

4400'  4480 

4560 

4640 

4720 

80 

543 

4800 

4880 

4960 

5040 

5120 

52001 52 ro 

5359 

5439 

5519 

80 

544 

5599 

5679 

5759 

5838 

5918 

5998! 6078 

6157 

6237 

6317 

80 

545 

6397 

6476 

6556 

6635 

6715 

6795 

6874 

6954 

7034 

7113 

80 

546 

7193 

7272 

7352 

7431 

7511 

7590 

7670 

7749 

7829 

7908 

79 

547 

7987 

8067 

8146 

8225 

8305 

8384 

8403 

8543 

8022 

8701 

79 

548 

87S1 

8860 

8939 

9018 

9097 

9177 

9266 

9335 

9414 

9493 

79 

549 
550 

95/2 

9651 
0442 

9731 
0521 

9810 

9889 
0678 

996^ 

..47 

.126 
0915 

.206 

.284 

79 
79 

740363 

0600 

0757 

0836 

0994 

1073 

551 

1152 

1230 

1.309 

1388 

1467 

1546 

1624 

1703 

1782 

1860 

79 

552 

1939 

2018 

2096 

2175 

2254 

2332 

2411 

2489 

2568 

2646 

79 

553 

2725 

2804 

2882 

2961 

3039 

3118 

3196 

3275 

.3353 

3431 

78 

554 

3510 

3588 

3667 

3745 

3823 

3902 

3980 

4058 

4136 

4215 

78 

555 

4293 

4371 

4449 

4528 

4606 

4684 

4762 

4840 

4919 

4997 

78 

556 

5075 

5153 

5231 

5309 

.5387 

5465 

5543 

5621 

5699 

6777 

78 

557 

6855 

5933 

6011 

6089 

6167 

6245 

6323 

6401 

6479 

6556 

78 

558 

6G34 

6712  6790 

6868 

6945 

7023 

7101 

7179 

7256 

7334 

78 

559 

560 

7412 

7489  7567 

7645 

8421 

7722 
8498 

7800 
8576 

7878 
8653 

7955 

8731 

8033 

8808 

8110 

8885 

78 
77 

748188 

82G6 

8.343 

561 

8963 

9040 

9118 

9195 

9272 

9350 

9427 

9504 

9582 

9659 

77 

502 

9736 

9814 

9891 

9968 

..45 

.123 

.200 

.277 

.3.54 

.431 

77 

563 

750508 

0586 

0663 

0740 

0817 

0894 

0971 

1048 

1125 

1202 

77 

564 

1279 

1356 

1433 

1510 

1.587 

1664 

1741 

1818 

1895' 1972 

77 

565 

2048 

2125 

2202 

2279 

2356 

2433 

2509 

2586 

2663  2740 

7? 

566 

2816 

2893 

2970 

3047 

3123 

3200 

3277 

3353 

3430  3506 

77 

567 

3583 

3660 

3736 

3813 

3889 

3966 

4042 

4119 

4195  4272 

77 

568 

4348 

4425 

4.501 

4578 

46,54 

4730 

4807 

4883 

4900  .5036  76  I 

569 
570 

5112 

5189 
5951 

5265 
6027 

5341 
6103 

.5417 

5494 
6256 

5570 
6332 

.5646 
6408 

.5722  5799 
6484  6.560 

76 
76 

755875 

6180 

671 

6636 

6712 

6788 

6864 

6940 

7016 

7092 

7168 

7244'  7.320 

76 

572 

7396 

7472 

7548 

7624 

7700 

7776 

7851 

7927 

8003  8079 

76 

573 

8155 

8230 

8306 

8382 

8458 

8533 

8609 

8685 

8761  8836 

76 

574 

8912 

8988 

9063 

9139 

9214 

9290 

9366 

9441 

9517,  9.592 

76 

575 

9668 

9743 

9819 

9894 

9970 

..45 

.121 

.196 

.272!  .347 

75 

576 

760422 

0498 

0573 

0649 

0724 

0799 

0875 

0950 

1025  1101 

76 

677 

1176 

1251 

1326 

140t4 

1477 

15.52 

1627 

1702 

1778, 1853 

75 

578 

1928 

2003 

2078 

2153 

2228 

2303 

2378 

2463 

2529  2604 

76 

579 

2679 

27.5412829 

2904-  2978 

30.53  3128 

3203 

3278  3353'  75 1 

_N^_ 

U   1  1   1  2  1  3  1  4  1  5  1  6  1  7  1  8  i  9  ■  i>. 

6  6. J 


10 


LOGARITHMS   OF   NUMBERS. 


N. 

J   0   |l|2|3i4|5|6|7|8|9|D.  1 

580 

763428. 3503 

3078,  3653 

3727 

3802:  3877,  395:^ 

4027 

4101 

75 

581 

4176 

4251 

4326 

4400 

4475 

455C 

462^^ 

4699 

4774 

4848 

75 

682 

4923 

4998 

5072 

5147 

5221 

5296 

537( 

5445 

5520 

5594 

75 

583 

5669 

5743 

5818 

5892 

5966 

0041 

6il£ 

6190  6264 

6338 

74 

584 

6413 

6487 

6562 

6636 

6710 

6785 

685JJ 

6933  7007 

7082 

74 

J85 

7156 

7230 

7304 

7379 

7453 

7527 

7601 

7675  7749 

7823 

74 

586 

7898 

7972 

8046 

8120 

8194 

8268 

8342 

841618490 

8564 

74 

587 

8638 

8712 

8786 

8860 

8934 

9008 

9082 

9156 

9230 

9303 

74 

588 

937'. 

9451 

9525 

9599 

9673 

9746 

9820 

9894 

9968 

.42 

74 

589 
590 

770115 

0189 

0263 
0999 

0336 

0410 
1146 

0484 
1220 

0557 
1293 

0631 

0705 
1440 

0778 
1514 

74 
74 

770852 

0926 

1073 

1367 

591 

1587 

1661 

1734 

1808 

1881 

1955 

2023 

2102 

21  r5 

2248 

73 

592 

2322 

2395 

2468 

2542 

2615 

2688 

2762 

2835 

2908 

2981 

73 

593 

3055 

3128 

3201 

3274 

3348 

3421 

3494 

3567 

3640 

3713 

73 

594 

3786 

3860 

3933 

4006 

4079 

4152 

4225 

4298 

4371 

4444 

73 

595 

4517 

4590 

4663 

4736 

4809 

4882 

4955 

5028 

5100 

5173 

73 

596 

5246 

5319 

5392 

5465 

5538 

5610 

1  5683 

5756 

5829 

5902 

73 

597 

5974 

6047 

6120 

6193 

6265 

6338 

1  6411 

6483 

6556 

6629 

73 

598 

6701 

6774 

6846 

6919 

6992 

7064 

7137 

7209 

7282 

7354 

73 

599 
600 

7427 

7499 
8224 

7572 
8296 

7644 

7717 

7789 
8513 

7862 
8585 

79.34 

8006 
8730 

8079 

8802 

72 

"72 

770151 

8368 

8441 

86.58 

601 

8874 

8947 

9019 

9091 

9163 

9236 

9308 

9380 

9452 

9524 

72 

602 

9596 

9669 

9741 

9813 

9885 

9957 

..29 

.101 

.173 

.245 

72 

603 

780317 

0389 

0461 

0533 

0605 

0677 

0749 

0821 

0893 

0965 

72 

604 

1037 

1109 

1181 

1253 

1324 

1396 

1468 

1.540 

1612 

1684 

72 

605 

1755 

1827 

1899 

1971 

2042 

2114 

2186 

2258 

2329 

2401 

72 

606 

2473 

2544 

2616 

2688 

2759 

2831 

2902 

2974 

3046 

3117 

72 

607 

3189 

3260 

3332!  3403 

3475 

3546 

3618 

3689 

3761 

3832 

71 

608 

3904 

3975 

4046,4118 

4189 

4261 

4332 

4403 

4475 

4.546 

71 

609 

4617 

4689 

4760 

4831 

4902 

4974 

5045 

5116 

5187 

5^59 

Jl 

610 

785330 

6401 

5472 

5543 

5615 

5686 

5757 

5828 

5899 

.5970 

71 

611 

6041 

6112 

6183 

6254 

6325 

6396 

6467 

6538 

6609 

6680 

71 

612 

6751 

6822 

6893  6964 

7035! 

7106 

7177 

7248 

7319 

7390 

71 

613 

7460 

7531 

7602 

7673 

7714 

7815 

7885 

7956 

80271  8098 

71 

614 

8168 

8239 

8310 

8381 

8451 

8522 

8593 

8663!  8734 

8804 

71 

615 

8875 

8946 

9016 

9087 

9157 

9228 

9299 

9369 

9440 

9510 

71 

616 

9581 

9651 

9722 

9792 

9863 

9933 

...4 

..74 

.144 

-.215 

70 

617 

790285 

0356 

0426 

0496 

0567 

0637 

0707  0778 

0848 

0918 

70 

618 

0988 

1059 

1129 

1199 

1269 

1340 

1410 

1480 

i:50 

1620 

70 

619 
620 

1691 

1761 

1831 
2532 

1901 
2602 

1971 
2672 

2041 

2111 
2812 

2181 

2882 

2252 
29?.2 

2322 
3022 

70 
"70 

792392 

2462 

2742 

621 

3092 

3162 

3231 

3301 

3371 

3441 

3511 

3581 

3651 

3721 

70 

622 

3790 

3860 

3930 

4000 

4070 

4139 

4209 

4279 

4349 

4418 

70 

623 

4488 

4558 

4627 

4697 

4767 

4836 

4906 

4976 

.5045 

5115 

70 

624 

5185 

5254 

5324 

5393 

5463' 

5532 

5602 

5672 

5741 

5811 

70 

625 

5880 

5949 

6019 

6088 

6158! 

6227 

6297 

6366 

6436 

6505 

69 

626 

6574 

6644 

67131  67821 

6852  6921 

6990 

7060 

7129 

7198 

69 

627 

7268 

7337 

7406  7475! 

7545^  7614 

7683 

7752 

7821 

7890 

69 

628 

7960 

8029 

8098 

8167 

8238 1 8305 

8374 

8443 

8513 

8582 

69 

629 
630 

8651 

8720 
9409 

8789 
9478 

8858 
9547 

8927  8996 
96 10  9685 

9065 
97.54 

9134 
9823 

9203 
9892 

9272 
9961 

69 
69 

799341 

631 

800029 

0098 

0167 

0236 

0305  0373 

0442 

0511 

0580 

0648 

69 

632 

0717 

0786 

0854 

0923 

0992,  1061 

1129 

1198 

1266 

1335 

69 

633 

1404 

1472 

1541 

1609| 

1678  1747 

1815 

1884 

1952 

2021 

€9 

634 

2089 

2158 

2226 

2295] 

2363  2432 

2500 

2568 

2637 

2705 

69 

635 

2774 

2842 

2910 

2979 

3047|3116 

3184 

3252 

3321 

3389 

68 

636 

3457 

3525 

3594 

3G62 

3730  3798 

38G7 

3935 

4003 

4071 

68 

637 

4139 

4208 

4276 

4344 

4412  4480 

4548 

4616 

4685 

4753 

68 

638 

4821 

4889! 

4957 

5025; 

5093  5lBl 

5229  52U7I 

5365 

.5433 

68 

039 

5501155691 

5637 

5705  57731  5841'  5908i  5976:  6044'  61 12'  68  | 

Jld 

0   |l|2i3|4l5|6|7|8|9|D.  1 

^^/),   5-6'^^ 


LOGARITHMS   OF  NUMBERS. 


-"• 

1   0 

\     ^ 

1  2  1  3 

4|5  |6|7|8|9|U.  1 

(540 

806180 

6248 

6316|  6384 

0451 

6519 

6587 

6055,  0723 

6790 

68 

041 

6858 

6926 

699417061 

7129 

7197 

7204 

7332 

7400 

7407 

68 

642 

7535 

7603 

7670  7738 

7800 

7873 

7941 

8008 

8076 

8143 

68 

043 

8211 

8279 

8346:8414 

8481 

8549 

80101  8084 

8751 

8818 

67 

644 

8886 

80  "iS 

9021 

9088 

9150 

9223 

92901  9358 

9425 

9492 

67 

645 

9560 

9627 

9094 

9702 

9829 

9896 

9904 

..31 

..98 

.105 

67 

646 

810233 

0300 

0367 

0434 

0501 

0569 

0036 

0703 

0770 

0837 

67 

647 

0904 

0971 

1039 

1100 

1173 

1240 

1307 

1374 

1441 

1508 

67 

648 

1575 

1642 

1709 

1770 

1843 

1910 

1977 

2044 

2111 

2178 

67 

649 
6-50 

2245 

2312 

2379 

2445 

2512 

2579 
3247 

2646 
3314 

2713 
3381 

2780 
3448 

2847 
3514 

67 

"07 

812913; 2980 

3047 

3114 

3181 

651 

3581 

3648 

3714 

3781 

3948 

3914 

3981 

4048 

4114 

4181 

67 

652 

4248 

4314 

4381 

4447 

4514 

4581 

4647 

4714 

4780 

4847 

67 

653 

4913 

4980 

5046 

5113 

5179 

5246 

5312 

5378 

5445 

.5511 

66 

654 

6578 

5644 

5711 

5777 

5843 

,5910 

5976 

0042 

6109 

6175 

66 

655 

6241 

6308 

6374 

0440 

0500 

m 

6639 

0705 

6771 

6838 

66 

656 

6904 

6970 

7036 

7102 

7109 

T3'01 

7307 

7433 

7499 

66 

657 

7565 

7631 

7698 

7704 

7830 

7896 

7962 

8028'  8094 

8160 

66 

658 

8226 

8292 

8358 

8424 

8490 

8550 

8622 

8088 

8754 

8820 

66 

659 
660 

8885 
819544 

8951 

9017 
9676 

9083 

9149 
9807 

9215 

9281 
9939 

9340 

...4 

9412 
..70 

9478 

66 
66 

9610 

9741 

9873 

.1.36 

661 

820201 

0267 

0333 

0399 

0404 

0530 

0595 

0001 

0727 

0792 

66 

662 

0858 

0924 

0989 

1055 

1120 

1180 

1251 

1317 

1382 

1448 

66 

663 

1514 

1579 

1645 

1710 

1775 

1841 

1906 

1972 

2037 

2103 

65 

664 

2168 

2233 

2299 

2304 

2430 

2495 

2560 

2020 

2091 

2756 

65 

665 

2822 

2887 

2952 

3018 

3083 

3148 

3213 

3279 

3344 

3409 

65 

666 

3474 

3539 

3605 

3070 

3735 

3800 

3865 

3930 

3996 

4061 

65 

667 

4126 

4191 

4256 

4321 

4380 

4451 

4516 

4581 

4646 

4711 

65 

668 

4776 

4841 

4906 

4971 

5030 

5101 

5166 

5231 

5296 

5361 

65 

669 
670 

5426 
826075 

5491 

5556 
6204 

5021 

5080 
6334 

5751 
0399 

5815 
6464 

5880 
0528 

5945 
6593 

6010 
6658 

65 
65 

6140 

6269 

671 

6723 

6787 

6852 

6917 

6981 

7046 

7111 

7175 

7240 

7305 

05 

672 

7369 

7434 

7499 

7563 

7628 

7092 

7757 

7821 

7886 

7951 

65 

673 

8015 

8080 

8144 

8209 

8273 

8338 

8402 

8467 

8531 

8595 

64 

674 

8660 

8724 

8789 

8853 

8918 

8982 

9046 

9111 

9175 

9239 

64 

675 

9304 

9368 

9432 

9497 

9561 

9025 

9090 

9754 

9818 

9882 

64 

676 

9947 

..11 

..75 

.139 

.204 

.208 

.332 

•396 

.460 

.525 

64 

677 

830589 

0653 

0717 

0781 

0845 

0909 

0973 

1037 

1102 

1166 

64 

678 

1230 

1294 

1358 

1422 

1486 

1550 

1614 

1678 

1742 

1806 

64 

679 

680 

1870 
832509 

1934 
2573 

1998 
2637 

2062 
2700 

2126 
2764 

2189 

2828 

2253 

2317 
2956 

2381 

2445 

64 
64 

2892 

3020 

3083 

681 

3147 

3211 

3275 

3338 

3402 

3460 

3530 

3593 

3657 

3721 

64 

682 

3784 

3848 

3912 

3975 

4039 

4103 

4166 

4230 

4294 

4357 

64 

683 

442! 

4484 

4548 

4611 

4675 

4739 

4802 

4866 

4929 

4993 

64 

684 

5050 

5120 

5183 

5247 

5310 

5373 

5437 

5500 

5564 

5627 

63 

685 

5691 

5754 

5817 

5881 

5944 

0007 

6071 

6134 

6197 

6201 

63 

686 

6324 

6387 

6451 

6514 

6577 

0041 

6704 

6767 

68.30 

6894 

63 

687 

6957 

7020 

7083 

7146 

7210 

7273 

7336 

7399 

7402 

7525 

63 

688 

7588 

7652  7715 

7778 

7841 

7904 

7907 

8030 

8093 

81.56 

63 

689 
690 

8219 
838849 

8282 

8345 

8408 
9038 

8471 

8534 
9104 

8597 

8660 
9289 

8723 
9352 

8786 

63 
63 

8912 

8975 

9101 

9227 

9415 

691 

9478 

9541 

9604 

9667 

9729 

9792 

9855 

9918 

9981 

..43 

63 

692 

840106 

0169 

0232 

0294 

0357 

0420 

0482 

0545 

0008 

0671 

63 

693 

0733 

0796  0859 

0921 

0984 

1040 

1109 

1172 

1234 

1297 

63 

694 

1359 

1422 

1485 

1547 

1610 

1672 

1735 

1797 

1800 

1922 

63 

695 

1985 

2047 

2110 

2172 

2235 

2297 

2300 

2422 

2484 

2547 

62 

696 

2609 

2672 

2734 

2796 

2859 

2921 

2983 

3046'  3108 

3170 

62 

697 

3233 

3295 

3357 

3420 

3482 

3544 

3000 

3669  3731 

3793 

62 

698 

3855 

3918 

3980 

4042 

4104 

4166 

4229 

4291  4353 

4415 

62 

699 

4477 

4539 

4601 

4604 

4726 

4788 

4850 

4912  4974  .5036'  6*i  1 

N.  1 

0   !l|2|3|4|6|6|7| 

_8  1 

.?»  1 

-iT! 

12 


LOGARITHMS   OF  NUMBERS. 


N. 

1   0   |l|2|3|4|oit>|V;8!9|D. 

700 

845098 

5160.5222 

5284 

534 6 1 5408 

.5470|  5532 

5.594  |5656i  621 

701 

5718 

5780,  5842 

5904 

5966 

6028 

609C 

6151 

6213 

6275 

62 

702 

6337 

6399 

6461 

6523 

6585 

6646 

6708 

6770 

6832 

6894 

62 

703 

6955 

7017 

7079 

7141 

7202 

7264 

732T> 

7388 

7449 

7511 

62 

704 

7573 

7634 

7696 

7758 

781.1' 7881 

7943 

8004 

8066 

8128 

62 

705 

8189 

8251 

8312 

8374 

843.V8497 

8559 

8620 

8682 

8743 

62 

706 

8805 

8866 

8928 

8989 

9051 

9112 

9174 

9235 

9297 

935S 

61 

707 

9419 

9.481 

9542 

9604 

9665 

9726 

9788 

9849 

9911 

9972 

61 

708 

850033 

0095 

0156 

0217 

0279 

0340 

0401 

0462 

0524 

0585 

61 

709 
710 

0646 

0707 
1320 

0769 

0830 
1442 

0891 
1503 

0952 
1564 

1014 

1075 

1136 

1747 

1197 

61 
61 

851258 

1381 

1625 

1686 

1809 

711 

1870 

1931 

1992 

2053 

2114 

2175 

2236 

2297 

2358 

2419 

61 

712 

2480 

2541 

2602 

2663 

2724 

2785 

2846 

2907 

2968 

3029 

61 

713 

3090 

3150 

3211 

3272 

3333 

3394 

3455 

3516 

.3577 

3637 

61 

7J4 

3898 

3759 

3820 

3881 

3941 

4002 

40(53 

4124 

4185 

4245 

61 

7i5 

4306 

4367 

4428 

4488 

4549 

4610 

4670 

4731 

4792 

4852 

61 

716 

4913 

4974 

5034 

5095 

5156 

5216 

.5277 

5337 

5398 

5459 

61 

717 

5519 

5580 

5640 

5701 

5761 

5822 

5882 

5943 

6003 

6064 

61 

718 

6124 

6185 

6245 

6306 

6366 

6427 

6487 

6.548 

6608 

6668 

60 

719 
720 

6729 

67891 6850 

6910 
7513 

6970 
7574 

~031 
7634 

7091 
7694 

7152 

7212 
7815 

7272 

7875 

60 
60 

857332 

7393 

7453 

7755 

721 

7935 

7995 

8056 

8116 

8176 

8236 

8297 

8357 

8417 

8477 

60 

722 

8537 

8597 

8657 

8718 

8778 

8838 

8898 

8958 

9018 

9078 

60 

723 

9138 

9198 

9258 

9318 

9379 

9439 

9499 

95.59 

9619 

9679 

60 

724 

9739 

9799 

9859 

9918 

9978 

.  .38 

..98 

.1.58 

.218 

.278 

60 

725 

860338 

0398 

0458 

0518 

0578 

0637 

0697 

0757 

0817 

0877 

60 

726 

0937 

0996  10561 

1116 

1176 

1236 

1295 

1355 

1415 

1475 

60 

727 

1534 

1594 

1654 

1714 

1773 

1833 

1893 

19.52 

2012 

2072 

60 

728 

2131 

2191 

2251 

2310 

2370 

24.30 

2489 

2549 

2608 

2668 

60 

729 
730 

2728 

2787 

2847 
3442 

2906 

2966 

3025 

3085 
3080 

3144 
3739 

3204 
3799 

3263 

60 
59 

863323 

3382 

3501 

3561 

3620 

3858 

731 

3917 

3977 

4036 

4096 

4155 

4214 

4274 

4333 

4392 

4452 

59 

732 

4511 

4570 

4630 

4689 

4748 

4808 

4867 

4926 

4985 

5045 

59 

733 

5104 

5163 

5222 

5282 

5341 

5400 

5459 

.5519 

5578 

5637 

59 

734 

5696 

5755 

5814 

5874 

5933 

5992 

6051 

6110 

6169 

6228 

59 

735 

6287 

6346 

6405 

6465 

6524 

6583 

6642 

6701 

6760 

6819 

59 

730 

6878 

6937 

6996 

7055 

7114 

7173 

7232 

7291 

7350 

7409 

59 

737 

7467 

7526 

7585 

7644 

7703 

7762 

7821 

7880 

7939 

7998 

59 

738 

8056 

8115 

8174 

8233 

8292 

8350 

8409 

8468 

8527 

8.586 

59 

739 
740 

8644 

8703 
9290 

8762 

8821 
9408 

8879 
9466 

89.38 
95/5" 

8997 
9584 

9056 

9114 
9701 

9173 
9760 

59 
59 

869232 

9349 

9642 

741 

9818 

9877 

9935 

9994 

..53 

.111 

.170 

.228 

.287 

.,345 

59 

742 

870404 

0462 

0521 

0579 

0638 

0696 

0755 

0813 

0872 

0930 

h8 

743 

0989 

1047 

1106 

1164 

1223 

1281 

1339 

1398 

14.56 

1515 

58 

744 

1573 

1631 

1690 

1748 

1806 

1865 

1923 

1981 

2040 

2098 

58 

745 

2156 

2215 

2273 

2331 

2389 

2448 

2506 

2564 

2622 

2681 

58 

746 

2739 

2797 

2855 

2913 

2972 

3030 

3088 

3146 

3204 

3262 

58 

747 

3321 

3379 

3437 

3495 

3553 

3611 

3669 

3727 

3785 

3844 

58 

748 

3902 

3960  4018 

4076 

4134 

4192 

4250 

4308 

4366 

4424 

^8 

749 
750 

4482 

4540  4598 

4656 

4714 
5293 

4772 
5351 

4830 
5409 

4888 

4945 

.5003 

.5.582 

58 

58 

875061 

5119 

5177 

5235 

,5466 

5524 

751 

5640 

5698 

5756 

5813 

5871 

5929 

.5987 

6045 

6102 

6160 

58 

752 

6218 

6276 

6333 

6391 

6449 

6507 

6564 

6622 

6680 

6737 

58 

753 

6795 

6853 

6910 

6968 

7026 

7083 

7141 

7199 

7256 

7314 

58 

754 

7371 

7429 

7487 

7544 

7602 

7659 

7717 

VVV4 

7832 

7889 

58 

755 

7947 

8004 

8062 

8119 

8177 

82.34 

8292 

8349 

8407 

8464 

57 

756 

8522 

8579 

8637 

8694 

8752 

8809 

8866 

8924 

8981 

9039 

57 

757 

9096 

9153 

9211 

9268 

9325 

9383 

9440 

9497 

9,555 

9612 

57 

758 

9669 

9726 

9784 

9841 

9898 

9956 

..13 

..70 

.127 

.185 

57 

759 

880242 

0299! 03561 04131 

0471 

0528 

0585 

0642 

0699  07.56 

57 

_Nd 

0   1  1  1  2  i  3  1  4  1  5  i  6  1  7  1  8  i  9  1  D.  ! 

LOGARITHMS   OF   NUMBERS. 


13 


N.  ' 

0   1  1  1  2  1  3  I  4  1  5  1  6  1  7  1  8  1  9  1  D.  ( 

760" 

880814  0871 

0928 

09851 1042 

1099  1156 

1213  1271!  1328  671 

761 

1385 

1442 

1499 

1556J  1613 

1670 

1727 

1784  1841 1 1898 

57 

762 

1955 

2012 

2069 

2126  2183 

2240 

2297 

23.54  241112468 

57 

763 

2525 

2581 

2638 

2695! 2752 

2809 

2866 

2923  2980  3037 

57 

76  i 

3093 

3150 

3207 

3264  3321 

3377 

3434 

.3491  3548  3605 

57 

765 

3661 

3718 

3775 

3832, 3888 

3945 

4002 

40.59  4115  4172  57  | 

766 

4229 

4285 

4342 

4399:4455 

4512 

4569 

4625  4682: 4739 

57 

767 

479') 

4852 

4909 

4965; 5022 

5078 

5135 

5192  5248 i  .5305 

57 

768 

5361 

5418 

5474 

5531 15587 

5644 

5700 

5757 

581315870 

57 

769 

5926 

5983 

6039 

6096  6152 

6209  6265 

6321 

6378!  6434 

56 

770 

886491 

6547 

6604 

6660  6716 

6773 

6829 

6885 

6942' 6998 

56 

771 

7054 

7111 

7167 

72231 7280 

7336 

7392 

7449 

7505: 7561 

56 

772 

7617 

7674 

7730 

77861  7842 

7898 

7955 

8011 

8067  8123 

56 

773 

8179 

8236 

8292 

8348! 8404 

8460 

8516 

8573 

8629  8685 

56 

774 

8741 

8797 

8853 

8909' 8965 

9021 

9077 

9134 

9190, 9246 

56 

775 

9302 

9358 

9414 

9470  9526 

9582 

9638 

9694 

97501 9806 

56 

776 

9862 

9918 

9974 

..30  ..86 

.141 

.197 

.253 

.309 

.365 

56 

777 

890421 

0477 

0533 

0589  0645 

0700 

0756 

0812 

0868 

0924 

56 

778 

0980 

1035 

1091 

1147  1203 

1259 

1314 

1370 

1426 

1482 

56 

779 

780 

1537 

1593 
2150 

1649 
2206 

17051 1760 
226212317 

1816 
2373 

1872 
2429 

1928 

2484 

1983 
2540 

2039 
2595 

56 
56 

892095 

781 

2651 

2707 

2762 

2818 

2873 

2929 

2985 

3040 

3096 

3151 

56 

782 

3207 

3262 

3318 

3373 

3429 

3484 

3540 

3595 

3651 

3706 

56 

783 

3762 

3817 

3873 

3928 

3984 

4039 

4094 

4150 

4205 

4261 

55 

784 

4316 

4371 

4427 

4482 

4538 

4593 

4648 

4704 

4759 

4814 

55 

785 

4870 

4925 

4980 

5036 

5091 

5146 

5201 

5257 

5312 

5367 

55 

786 

5423 

5478 

5533 

5588 

5644 

5699 

5754 

5809 

.5864 

5920 

55 

787 

5975 

6030 

6085 

6140 

6195 

6251 

6306 

6361 

6416 

6471 

55 

788 

6526 

6581 

6636 

6692 

6747 

6802 

6857 

6912  6967 

7022 

55 

789 
790 

7077 

7132 
7682 

7187 
7737 

7242 
7792 

7297 

7847 

7352 
7908 

7407 
7957 

7462 
8012 

7517 
8067 

7572 
8122 

55 
55 

897627 

791 

8176 

8231 

8286 

8341 

8396 

8451 

8506 

8561 

8615 

8670 

55 

792 

8725 

8780 

8835 

8890 

8944 

8999 

9054 

9109 

9164 

9218 

55 

793 

9273;  9328 

9383 

9437 

9492 

9547 

9G02 

9656 

9711 

9766 

55 

794 

982110875 

9930 

9985 

..39 

..94 

.149 

.203 

.258 

.312 

55 

795 

900367  0422 

0476 

0531 

0586 

0640  0695 

0749 

0804 

0859 

55 

79G 

0913i 0968 

1022 

1077 

1131 

1186 

1240 

1295 

1349 

1404 

55 

797 

1458 

1513 

1567 

1622 

1676 

1731 

1785 

1840 

1894 

1948 

54 

798 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

54 

799 
800 

2547 

2601 
3144 

2655 
3199 

2710 
3253 

2764 
3307 

2818 
3361 

2873 
3416 

2927 
3470 

2981 
3524 

3036 
3578 

54 
54 

903090 

801 

3633 

3687 

3741 

3795 

3849 

3904 

3958 

4012 

4066 

4120 

54 

802 

4174 

4229 

4283 

4337 

4391 

4445 

4499 

4553 

4607 

4661 

54 

803 

4716 

4770 

4824 

4878 

4932 

4986 

5040 

5094 

5148 

.5202 

54 

804 

5256 

5310 

5364 

5418 

5472 

5526 

5580 

5634 

5688 

5742 

54 

805 

5796 

5850 

5904 

5958 

6012 

6066 

6119 

6173 

6227 

6281 

54 

806 

6335 

6389 

6443 

6497 

6551 

6604 

6658 

6712 

6766 

6820 

54 

907 

6874 

6927 

6981 

7035 

7089 

7143 
7680 

7196 

7250 

7304 

7358 

54 

808 

7411' 7405 

7519 

7573 

7626 

7734 

7787 

7841 

7895 

54 

809 

7949  8002 

8056 

8110 

8163 

8217 

8270 

8324 

8378 

8431 

54 

810 

908485  8539 

8592 

8646 

8699 

8753 

8807 

8860 

89141 8967 

54 

811 

9021  9074 

9128 

9181 

9235 

9289 

9342 

9396 

9449 

9503 

54 

812 

955619610 

9663 

9716 

9770 

9823 

9877 

9930 

9984 

..37 

53 

813 

910091 

0144 

0197 

0251 

0304 

0358 

0411 

0464 

0518 

0.571 

53 

814 

0624 

0678 

0731 

0784 

0838 

0891 

l0944 

0998 

1051 

1104 

53 

815 

1158 

1211 

1264 

1317 

1371 

1424 

1  1477 

15.30 

1584 

1637 

53 

816 

1690 

1743 

1797 

1850 

1903 

1956 

I2OO9 

2063 

2116 

2169 

53 

817 

2222 

2275 

2328 

2381 

2435 

2488 

12541 

2594 

2647 

2700 

53 

818 

2753 

2806 

!  2859 

2913 

2966 

3019 

3072 

3125 

3178 

3231 

53 

819 

3284 

3337 

1  3390 

34431 3496 

3549 

1  3602 

3655 

3708' 3761 

53 

N  _ 

1   0   1  1  1  *  1  3  1  4  1  5  1  6  I  7  1  8  1  U  1  D.l 

14 


LOGARITHMS    OF   NUMBERS. 


N. 

0   1  1  !  2  1  3  1  4  1  5  1  6  1  7  I  8  1  9  1  D. 

820 

913814,3867 

3920 

3973 

4026 

4079 

4132 

4184 

4237 

4290 

63 

821 

43431  4396 

4449 

4502 

4555 

4608 

4660 

4713 

4766 

4819 

63 

822 

4872  4925 

4977 

6030 

.5083 

5136 

5189 

.5241 

6294 

.5347 

53 

823 

5400 

5453 

5505 

6558 

.5611 

.5664 

,5716 

6769 

.5822 

5875 

63 

824 

5927 

5980 

6033 

6085 

61.38 

6191 

6243 

6296 

6349 

6401 

53 

825 

6t54 

6507 

6559 

6612 

6664 

6717 

6770 

6822 

6875 

6927 

53 

S2B 

6980, 7033 

7085 

71.38 

7190 

7243 

7295 

7348 

7400 

7453 

53 

827 

750C  7558 

7611 

7663 

7716 

7768 

7820 

7873 

7925 

7978 

52 

828 

80301 8083 

8135 

8188 

8240 

8293 

8345 

8397 

8450 

8.502 

62 

829 
830 

8555 

8607 

8659 

8712 

8764 
928- 

8816 
9.340 

8869 
9392 

8921 
9444 

8973 
9496 

9026 
9.549 

62 
62 

919078 

9130 

9183 

9235 

831 

9001 

9653 

9706 

9758 

9810 

9862 

9914 

9967 

..19 

.71 

62 

832 

920123 

0176 

0228 

0280 

0332 

0384 

0436 

0489 

0541 

0593 

62 

833 

0645 

0697 

0749 

0801 

0853 

0900 

09.58 

1010 

1062 

1114 

52 

834 

1166 

1218 

1270 

1.322 

1374 

1426 

1478 

1.530 

1.582 

1634 

52 

835 

1686 

1738 

1790 

1842 

1894 

1946 

1998 

2060 

2102 

2154 

62 

836 

2206 

2258 

2310 

2362 

2414 

2466 

25!  8 

2570 

2622 

2674 

62 

837 

2725 

2777 

2829 

2881 

2933 

2985 

3037 

3089 

3140 

3192 

62 

838 

3244 

3296 

3348 

3399 

3451 

3503 

3555 

3607 

3658 

3710 

62 

839 
840 

3762 
924279 

3814 

3865 
4383 

3917 
4434 

3969 

4021 

4072 

4124 
4641 

4176 

4228 
4744 

52 
62 

4331 

4486 

4538 

4589 

4693 

841 

4796 

4848 

4899 

4951 

5003 

5054 

5106 

5157 

5209 

6261 

62 

842 

6312 

6364 

5415 

6467 

.5518 

.5570 

5621 

6673 

6725 

6776 

52 

843 

5828 

5879  5931 

5982 

6034 

6085 

6137 

6188 

6240 

6291 

51 

844 

6342 

6394 

6445 

6497 

6548 

6600 

6651 

6702 

67.54 

6805 

61 

845 

6857 

6908 

6959 

7011 

7062 

7114 

7165 

7216 

7268 

7319 

51 

846 

7370 

7422 

7473 

7524 

7576 

7627 

7678 

7730 

7781 

7832 

61 

847 

7883 

7935 

7986 

8037 

8088 

8140 

8191 

8242 

8293 

8345 

61 

848 

8396 

8447 

8498 

8549 

8601 

8652 

8703 

8764 

8805 

8867 

51 

849 
850 

8908 
9294 1 9 

8959 
9470 

9010 

9061 

9112 

9163 

9215 
9725 

9266 

9317 

9368 

61 
51 

9521 

957^ 

9623 

9674 

9776 

9827 

9879 

851 

9930 

9981 

..32 

..83 

.134 

.185 

.236 

.287 

.338 

.389 

61 

852 

930440 

0491 

0542 

0592 

0643 

0694 

0745 

0796 

0847 

0898 

61 

853 

0949 

1000 

1051 

1102 

11.53 

1204 

12.54 

1305 

1356 

1407 

51 

854 

1458 

1509 

1560 

1610 

1661 

1712 

1763 

1814 

1865 

1915 

51 

855 

1966 

2017 

2068 

2118 

2169 

2220 

2271 

2322 

2372 

2423 

51 

856 

2474 

2524 

2575 

2626 

2677 

2727 

2778 

2829 

2879 

2930 

61 

857 

2981 

3031 

3082 

3133 

3183 

3234 

3285 

3335 

3386 

3437 

61 

858 

3487 

3538 

3589 

3639 

3690 

3740 

3791 

3841 

3892 

3943 

51 

859 
860 

3993 

4044 

4094 

4145 
4650 

4195 
4700 

4246 
4751 

4296 

4347 

4852 

4397 

4448 

61 
60 

934498 

4549 

4599 

4801 

4902 

49.53 

801 

6003 

5054 

5104 

5154 

6205 

5255 

.5.306 

5356 

5406 

.5457 

60 

862 

5507 

5558 

5608 

5658 

5709 

5759 

6809 

6860 

5910 

5960 

50 

863 

6011 

6061 

6111 

0162 

6212 

6262 

6313 

6363 

6413 

6463 

60 

864 

6514 

6564 

6614 

6665 

6715 

6765 

6815 

6865 

6916 

6966 

60 

885 

7016 

7066 

7117 

7167 

7217 

7267 

7317 

7367 

7418 

7468 

60 

866 

7518 

7568 

7618 

7668 

7718 

7769 

7819 

7869 

7919 

7969 

60 

867 

8019 

8069 

8119 

8169 

8219 

8269 

8320 

8370 

8420 

8470 

60 

868 

8520 

8570 

8620 

8670 

8720 

8770 

8820 

8870 

8920 

8970 

60 

1869 
870 

9020 
939519 

9070 
9569 

9120 

9170 
9669 

9220 

9270 
9769 

9320 
9819 

9369 

9419 

9918 

9469 

60 
50 

9619 

9719 

9869 

9968 

871 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

50 

872 

0516 

0566 

0616 

0666 

0716 

0765 

0815 

0865 

0915 

0964 

50 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

50 

874 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1869 

1909 

1958 

50 

875 

2008 

2058 

2107 

2157 

2207 

2256 

2306 

2355 

2405 

2455 

50 

876 

2504 

2554 

2603 

2663 

2702 

2752 

2801 

2851 

2901 

2950 

50 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

344  5 

49 

878 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

49 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

49 

I     1     I    2     I    3 


1    7     I    8     '9     (  D.  1 


LOGARITHMS   OF   NUMBERS. 


15 


880" 

0   |l|2|3|4|5|6|7|8|9lD.  1 

944483  45321 

4581  46311 

4680 

4729 

4779 

482814877 

4927 

49 

881 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

.5321  .5370 

,5419 

49 

882 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

5862 

5912 

49 

883 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

49 

884 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

49 

885 

6943  6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

49 

886 

7434  7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875 

49 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8364 

49 

888 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

88,53 

49 

889 
890 

8902 

8951 

8999 

9048 
9536 

9097 

9146 
9634 

9195 
9683 

9244 
9731 

9292 

9780 

9,341 
9829 

49 
'49 

949390 

9439 

9488 

9585 

891 

9878 

9926 

9975 

..24 

..73 

.121 

.170 

.219 

.207 

.316 

49 

892 

950365 

0414 

0462 

0511 

0560 

0608 

0057 

0706 

0754 

0803 

49 

893 

0851 

0900 

0949 

0997 

1046 

1095 

1143 

1192 

1240 

12«9 

49 

894 

1338 

1386 

1435 

1483 

1532 

1.580 

1629 

1677 

1726 

1775 

19 

805 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

48 

896 

2308 

2356 

2405 

2453 

2502 

2550 

2599 

2647 

2696 

2744 

48 

89? 

2792 

2841 

2889 

2938 

2986 

3034 

3083 

3131 

3180 

3228 

48 

898 

3276 

3325 

3373 

3421 

3470 

.3518 

3566 

3615 

3663 

3711 

48 

899 
900 

3760 

3808 
4291 

3856 

3905 

4387 

3953 
4435 

4001 

4049 
4532 

4098 
4580 

4146 
4628 

4194 
4677 

48 
"48 

954243 

4339 

4484 

901 

4725 

4773 

4821 

4869 

4918 

4966 

.5014 

5062 

5110 

51.58 

48 

902 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

5592 

5640 

48 

903 

5688  57361 

5784 

5832 

5880 

5928 

.5976 

6024 

6072 

6120 

48 

904 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

65.53 

6601 

48 

305 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

48 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

48 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

48 

908 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

48 

909 
910 

8564 

8612 

8659 

8707 

8755 
9232 

8803 
9280 

8850 
9328 

8898 
9375 

8946 
9423 

8994 

48 
48 

959041 

9089 

9137 

9185 

9471 

911 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

98.52 

9900 

9947 

48 

912 

9995 

..42 

..90 

.138 

.185 

.2.33 

.280 

.328 

.376 

.423 

48 

913 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

48 

914 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

47 

915 

1421 

1469 

1516 

1563 

1611 

16.58 

1V06 

1753 

1801 

1848 

47 

916 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

47 

917 

2369 

2417 

2464 

2511 

2559 

2606 

2653 

2701 

2748 

2795 

47 

918 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

47 

919 
920 

3316 

3363 
3835 

3410 

3882 

3457 
3929 

3504 
3977 

3552 
4024 

3.'^99 
4071 

3646 

4118 

3693 
4165 

3741 

4212 

47 
47 

963788 

921 

42b0 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684 

47 

922 

4731 

4778 

4825 

4872 

4919 

4966 

.5013 

5061 

5108 

51.55 

47 

923 

5202 

5249 

5296 

5343 

5390 

.5437 

5484 

5531 

5578 

5625 

47 

924 

5672 

5719 

5766 

5813 

.5860 

5907 

5954 

6001 

6048 

6095 

47 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

47 

926 

6611 

6658 

6705 

6752 

679^^ 

6845 

6892 

6939 

6986 

7033 

47 

927 

7080 

7127 

7173 

7220 

7267 

"314 

7361 

7408 

74,54 

7,501 

47 

928 

7548 

7595 

7642 

7688 

7735 

/782 

7829 

7875 

7922 

7969 

47 

929 
930 

8016 

8062 
8530 

8109 
8576 

8156 

8203 
86/0 

8249 
8716 

8296 
8763 

8343 

8810 

8390 
8856 

8436 

8903 

47 
47 

968483 

8623 

931 

8950 

8996 

9043 

9090 

91.36 

9183 

9229 

9276 

9323 

9369 

47 

932 

9U6 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

47 

933 

9882 

9928 

9975 

..21 

..68 

.114 

.101 

207 

.2.54 

.300 

47 

934 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

46 

935 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

40 

936 

1276 

1322 

1369 

1415 

1461 

1508 

1.5.54 

1601 

1647 

1693 

46 

937 

1740 

1786 

1832 

187S  1925 

1971 

2018 

2064 

2110 

21.57 

46 

938 

2203 

2249 

22951  23421  2388 

2434 

2481 

2527 

2573 

2619 

46 

939 

26661  2712'  2758'  2804'  2851 

2897'  2943 

2989' 30.35' 3082'  46  | 

"nT 

1   0 

1  1 

1  2 

3 

1  4  1  5  1  6  !  7  1  8  1  9  1  D.  j 

16 


LOGARITHMS   OF  NUMBERS. 


N. 

0   |l|2j3|4|5|6|7|8|9|D.  1 

940 

973128 

3174 

3220 

3266 

33131  3359  3405 

3451  3497 

3543 

46 

941 

3590 

3636 

3682 

3728 

3774  3820  3866 

3913 

3959 

4005 

46 

942 

4051 

4097 

4143 

4189 

4235 

4281  4327 

4374 

4420 

4466 

46 

943 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

48.34 

4880 

4926 

46 

944 

4972 

5018 

5064 

5110 

5156 

5202 

5248 

5294 

.5.340 

5386 

46 

945 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

46 

946 

5891 

5937 

5983 

6029 

6075 

6121 

6167 

6212 

62.58 

6304 

46 

947 

6350 

6396  6442 

6488 

6533 

6579 

6625 

6671 

6717 

6763 

46 

948 

6808 

6854  6900 

6946 

6992 

7037 

7083 

7129 

7175 

7220 

46 

949 
950 

7266 
977724 

7312 

7358 

7815 

7403 

7449 
7906 

7495 

7541 
7998 

7586 

7632 

8089 

7678 
81.35 

46 
"46 

7769 

7861 

7952 

8043 

951 

8181 

8226 

8272 

8317 

8363 

8409 

8454 

8500 

8546 

8591 

46 

952 

8637 

8683 

8728 

8774 

8819 

8865 

8911 

8956 

9002 

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46 

953 

9093 

9138 

9184 

9230 

9275  9321 

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9412  9457 

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46 

954 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

45 

956 

0458 

0503 

0549 

0594 

0640 

0685 

0730 

0776 

0821 

0867 

45 

957 

0912 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1275 

1320 

45 

958 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

45 

959 
960 

1819 
982271 

1864 
2316 

1909 
2362 

1954 
2407 

2000 

2045 

2090 

2135 

2181 

2226 
2678 

45 
45 

2452 

2497 

2543 

2588 

2633 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

45 

962 

3175 

32201  32651  3310 

3356 

3401 

3446 

3491 

3536 

3581 

45 

963 

3626 

36711371613762 

3807 

3852 

3897 

3942 

3987 

4032 

45 

961 

4077 

41221416714212 

4257 

4302 

4347 

4392 

4437 

4^182 

45 

965 

4527 

4572 

4617 

4062 

4707 

4752 

4797 

4842 

4887 

4932 

45 

966 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

5292 

5337 

5382 

45 

967 

5426 

5471 

5516 

5561 

5606 

5651 

5696 

5741 

5786 

5830 

45 

968 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

45 

969 
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6324 

6369 
6817 

6413 

6458 
6906 

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6548 
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6593 

6637 
7085 

6682 

6727 

45 
45 

986772 

6861 

7040 

7130 

7175 

971 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7.532 

7577 

7622 

45 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

4,^ 

973 

8113 

8157 

8202 

8247 

8291 

83.36 

8381 

8425 

8470 

8514 

45 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

45 

975 

9005 

9049 

9094 

9138 

9183 

9227 

9272 

9316 

9.361 

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45 

976 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9701 

9806 

9850 

44 

977 

9895 

9939 

9983 

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..72 

.117 

.161 

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.294 

44 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

44 

979 
984) 

0783 
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0827 

0871 
1315 

0916 
1359 

0960 

1004 
1448 

1049 

1093 

1137 
1580 

1182 
1625 

44 

44 

1270 

1403 

1492 

1536 

981 

1669 

1713 

1758 

1802 

1846 

1890 

1935 

1979 

202'< 

2067 

44 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2.509 

44 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

44 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

44 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833 

44 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

44 

987 

4317 

4361 

4405 

4449 

4493 

4.537 

4.581 

4625 

4669 

4713 

44 

988 

4757 

4801 

4845 

4889 

4933 

4977 

5021 

.5065 

5108 

5152 

44 

989 
990 

5196 

5240 
5679 

5284 

5328 

.5372 

5416 
5854 

5460 

5504 
5942 

5547 

5591 
6030 

44 
44 

995635 

5723 

5767 

.5811 

.5898 

5986 

991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

44 

992 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

44 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

44 

994 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

44 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

44 

997 

8695 

8739 

8782 

8826 

8869 

8913 

89.56 

9000 

9043 

9087 

44 

998 

9131 

9174 

9218 

926] 

9305 

9348 

9392 

9435 

9479 

9.522 

44 

999 

9565 

9609 

9652 

9606 

9739 

9783 

9826 

987019913 

9957 

43 

"157 

1   0   1  1  1  2  I  3  1  4  1  5  1  6  1  7  1  8  1  9  1  D.  1 

^dr.  ?/-/n^- 


LOGARITHMIC 


SINES    AND    TANGENTS, 


EVERY  DEGREE  AND  MINUTE 


OF   THE   QUADRANT. 


The  minutes  in  the  left-hand  column  of  each  page,  increas- 
ing downward,  belong  to  the  degrees  at  the  top ;  and  those 
increasing  upward,  in  the  right-hand  column,  belong  to  the 
degrees  below. 


<^/.^fC^^^^  - 


5?^ 


/9??f^ib^-''/kJ^ 


18 


0°.      LOGAKITHMIC 


M 

1   Sine   1   D. 

Cosine   1  D. 

Tang.   1   U. 

Cotang. 

!_ 

~o" 

0.000000 

, 

10.000000    1 

0.000000 

Inniiiie. 

60 
59 

1 

6.403726 

501717 

000000  00 

6.463726 

501717 

13.536274 

2 

764756 

293485 

000000 

00 

764756 

293483 

235244 

58 

3 

940817 

208231 

000000 

00 

940847 

208231 

059153 

57 

4 

7.065786 

161517 

000000 

00 

7.065786 

161517 

12.934214 

56 

6 

162696 

131968 

000000 

00 

162696 

131969 

837304 

55 

6 

241877 

111575 

9.999999 

01 

241878 

111578 

758122 

54 

7 

308824 

96653 

999999 

01 

308825 

99653 

691175 

53 

8 

366816 

85254 

999999 

01 

366817 

85254 

633183 

52 

9 

417968 

76263 

999999 

01 

417970 

76263 

582030 

51 

10 
11 

463725 
7.505118 

68988 

999998 

01 
01 

463727 

68988 

536273 
12.494880 

50 
49 

62981 

9.999998 

7.5Col20 

62981 

12 

542906 

57936 

999997 

01 

542909 

57933 

457091 

48 

13 

577668 

53641 

999997 

01 

677672 

63612 

422328 

47 

14 

609853 

49938 

999996 

01 

609857 

49939 

390143 

46 

15 

039816 

46714 

999996 

01 

639820 

46715 

360180 

45 

16 

667845 

43881 

999995 

01 

667849 

43882 

332151 

44 

17 

694173 

41372 

999995 

01 

694179 

41373 

30.5821 

43 

18 

718997 

39135 

999994 

01 

719003 

39136 

280997 

42 

19 

742477 

37127 

999993 

01 

742484 

37128 

257c 16  41 

20 
21 

764754 
7.785943 

353.15 

999993 
9.999992 

01 
01 

764761 
7.785951 

35136 
33673 

235239'  40 

33672 

12.214049 

39 

22 

806146 

32175 

999991 

01 

806155 

32176 

19.3845 

38 

23 

825451 

30805 

999990 

01 

825460 

30806 

174540 

37 

24 

843934 

29547 

999989 

02 

843944 

29549 

156056 

36 

25 

861662 

28388 

999988 

02 

861674 

28390 

138326 

35 

26 

878695 

27317 

999988 

02 

878708 

27318 

121292 

34 

27 

895085 

26323 

999987 

02 

895099 

2C325 

104901 

33 

28 

910879 

25399 

999986 

02 

910894 

25401 

089106 

32 

29 

926119 

24538 

999985 

02 

926134 

24'>40 

073866 

31 

30 
31 

940842 
7.955082 

23733 

999983 

02 

02 

940858 
7.955100 

23735 
22981 

059142 

30 

29 

22980 

9.999982 

12.044900 

32 

968870 

22273 

999981 

02 

968889 

22275 

031111 

28 

33 

982233 

21608 

999980 

02 

982253 

21610 

017747 

27 

•SA 

995198 

20981 

999979 

02 

995219 

2)983 

004781 

26 

35 

8.007787 

20390 

999977 

02 

8.007809 

2.)392 

11.992191 

25 

30 

020021 

19831 

999976 

02 

020045 

19883 

979955 

24 

37 

031919 

19302 

999975 

02 

031945 

19305 

S68U55 

23 

38 

043501 

18801 

999973 

02 

043527 

18803 

956473 

22 

39 

054781 

18325 

999972 

02 

054809 

18327 

945191 

21 

40 

065776 

17872 

999971 

02 

065806 

17874 

934194 

20 

41 

8.076500 

17441 

9.999969 

02 

8.076531 

17444 

11.923469 

19 

42 

086965 

17031 

999968 

02 

086997 

17034 

913003 

18 

43 

097183 

16639 

999966 

02 

097217 

16642 

902783 

17 

44 

107167 

16265 

999964 

03 

107202 

16268 

892797 

16 

45 

116926 

15908 

999963 

03 

116963 

15910 

883037 

15 

46 

126471 

15566 

999961 

03 

126510 

15568 

873490 

14 

47 

135810 

15238 

999959 

03 

135851 

15241 

864149 

13 

48 

144953 

14924 

999958 

03 

144996 

14927 

855004 

12 

49 

153907 

14622 

999956 

03 

153952 

14627 

846048 

11 

50 
51 

162681 
8.171280 

14333 
14054 

999954 

03 
03 

162727 

14336 
14057 

837273 

11.828672 

10 
9 

9.999952 

9.171328 

52 

179713 

13786 

999950 

03 

179763 

13790 

820237 

8 

53 

187985 

13529 

999948 

03 

188036 

13532 

811964 

7 

54 

196102 

13280 

999946 

03 

196156 

13284 

803844 

6 

55 

204070 

13041 

999944 

03 

204126 

13044 

V95874 

5 

56 

211895 

12810 

999942 

04 

211953 

12814 

7fe8047 

4 

57 

219581 

12587 

999940 

04 

219641 

12590 

780359 

3 

58 

227134 

12372 

999938 

04 

227195 

12376 

772803 

2 

59 

234557 

12164 

999936 

04 

234621 

12168 

765379 

1 

60 

241855 

11963 

9999341  04 

241921 

11967 

75S07S 

0 

3 

Corvine  j 

1   «"•«   1 

Cutang.  j 

1    Tang.   j  Ki.  | 

SINES  AND  TANGENTS.      1°. 


19 


Ml 

Sine   1   D. 

Cosine   |  D. 

Tang.   {   D.   | 

Ootnng.   i 

0 

8.241855 

11963 

9.999934 

04 

8.241921 

11967 

11.758079  60 

1 

249033 

11768 

999932 

04 

249102 

11772 

750898  59 

2 

256094 

11.580 

999929 

04 

256165 

11584 

743835  58 

3 

263042 

11398 

999927 

04 

263115 

11402 

736H85  57 

4 

269881 

11221 

999925 

04 

269956 

11225 

730044  56 

5 

276614 

11050 

999922 

04 

276691 

11054 

723309 

55 

6 

283243 

10883 

999920 

04 

283323 

10887 

716677 

54 

7 

289773 

10721 

999918 

04 

289850 

10726 

710144 

53 

8 

296207 

10565 

999915 

04 

296292 

10570 

703708 

52 

9 

302546 

10413 

999913 

04 

302034 

10418 

697366 

51 

10 
11 

308794 
8.314954 

10266 

999910 
9.999907 

04 
04 

308884 
8.315046 

10270 

691116 

50 
49 

10122 

10126 

11.684954 

12 

321027 

9982 

999905 

04 

321122 

9987 

678878 

48 

13 

327016 

9847 

999902 

04 

327114 

9851 

672886 

47 

14 

332924 

9714 

999899 

05 

333025 

9719 

666975 

46 

15 

338753 

9586 

999897 

05 

338856 

9590 

661144 

45 

16 

344504 

9460 

999894 

05 

344610 

9465 

655390 

44 

17 

350181 

9338 

999891 

05 

350289 

9343 

649711 

43 

18 

355783 

9219 

999888 

05 

355895 

9224 

644105 

42 

19 

361315 

9103 

999885 

05 

361430 

9108 

638570 

41 

20 
21 

366777 

8990 

999882 

05 
05 

366895 
8.372292 

8995 

633105 

40 
39 

8.372171 

8880 

9.999879 

8885 

11.627708 

22 

377499 

8772 

999876 

05 

377622 

8777 

622378 

38 

23 

382762 

8667 

999873 

05 

382889 

8672 

617111 

37 

24 

387962 

8564 

999870 

05 

388092 

8570 

611908 

36 

25 

393101 

8464 

999867 

05 

393234 

8470 

006766 

35 

26 

398179 

8366 

999864 

05 

398315 

8371 

601685 

34 

27 

403199 

8271 

999861 

05 

403338 

8276 

596662 

33 

28 

408161 

8177 

999858 

05 

408304 

8182 

591696 

32 

29 

413068 

8086 

939854 

05 

413213 

8091 

586787 

31 

30 

417919 

7996 

999851 

06 

418068 

8002 

581932 

30 

31 

8.422717 

7909 

9.999848 

06 

8.422869 

7914 

11.577131 

29 

32 

427462 

7823 

999844 

06 

427618 

7830 

572382 

28 

33 

432156 

7740 

999841 

06 

432315 

7745 

567685 

27 

34 

436800 

7657 

999838 

06 

436962 

7663 

663038 

26 

35 

441394 

7577 

999834 

06 

441560 

7583 

658440 

25 

36 

445941 

7499 

999831 

06 

446110 

7505 

.553890 

24 

37 

450440 

7422 

999827 

06 

450613 

7428 

549387 

23 

38 

454893 

7346 

999823 

06 

455070 

7352 

644930 

22 

39 

459301 

7273 

999820 

06 

459481 

7279 

540519 

21 

40 

463665 

7200 

999816 

06 

463849 

7206 

536151 

20 

41 

8.467985 

7129 

9.999812 

06 

8.468172 

7135 

11.531828 

19 

42 

472263 

7060 

999809 

06 

472454 

7066 

527546 

18 

43 

478498 

6991 

999805 

06 

476693 

6998 

523307 

17 

44 

480693 

6924 

999801 

06 

480892 

6931 

619108 

16 

45 

484848 

6859 

999797 

07 

485050 

6865 

614950 

15 

46 

488963 

6794 

999793 

07 

489170 

6801 

610830 

14 

47 

493040 

6731 

999790 

07 

493250 

6738 

606750 

13 

48 

497078 

6669 

999788 

07 

497293 

6676 

502707 

12 

49 

601080 

6608 

999782 

07 

501298 

6615 

498702 

11 

60 

605045 

6548 

999778 

07 

505267 

6555 

494733 

10 

51 

8.508974 

6489 

9.999774 

07 

8.. 509200 

6496 

11.490800 

9 

62 

612867 

6431 

999769 

07 

613098 

6439 

486902 

Sj 

53 

516726 

6375 

999765 

07 

616961 

6382 

483039 

7 

64 

620551 

6319 

999701 

07 

520790 

6326 

479210 

6 

55 

524343 

6264 

999757 

07 

624586 

6272 

475414 

5 

56 

628102 

6211 

999753 

07 

628349 

6218 

471651 

4 

67 

631828 

6158 

999748 

07 

532080 

6165 

467920 

3 

58 

635523 

6106 

999744 

07 

535779 

6113 

464221 

2 

59 

639186 

6055 

999740 

07 

539447 

6062 

460553 

1| 

60^ 

542819!  6004 

999735' 07 

543081'  6012 

456916 

ol 

CO...U.  j 

1   Sine   1 

1  Cotang.   1 

1   Tang 

m 

83° 


20 


2°.      LOGARITHMIC 


,  M. 

Sine 

— a-- 

Cosine   |  D. 

1   Tane.   j    D. 

1   rotane.   1   ] 

0 

8.542819 

6004 

9.999735 

07 

18.543084 

6012 

11.456916 

60 

1 

546422 

5955 

999731 

07 

646691 

.5962 

453309 

59 

2 

549995 

5906 

999726 

07 

550268 

5914 

449732 

58 

3 

553539 

5858 

999722 

08 

653817 

5866 

446183 

57 

4 

557054 

.5811 

999717 

08 

557336 

5819 

442664 

56 

5 

560540 

5765 

999713 

08 

560828 

5773 

439172 

65 

6 

563999 

5719 

999708 

08 

564291 

5727 

435709 

54 

7 

567431 

5674 

999704 

08 

567727 

5682 

432273 

63 

8 

570836 

5630 

999699 

08 

571137 

5638 

428863 

52 

9 

574214 

5587 

9i)9694 

08 

574520 

5595 

425480 

51 

10 
11 

577566 

8.580892 

5544 

999689 

08 
08 

677877 
8.581208 

5552 

422123 

50 

49 

5502 

9.999685 

5510 

11.418792 

12 

584193 

5460 

999680 

08 

584514 

,5468 

415486 

48 

13 

687469 

5419 

999675 

08 

687795 

.5427 

412205 

47 

14 

590721 

5379 

999670 

08 

591051 

5387 

408919 

46 

15 

593948 

5339 

999665 

08 

594283 

5347 

405717 

45 

16 

597152 

5300 

999660 

08 

597492 

5.308 

402508  44 | 

17 

600332 

5261 

999655 

08 

600677 

5270 

399323 

43 

18 

603489 

5223 

999650 

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603839 

6232 

396161 

42 

19 

606623 

5186 

999645 

09 

606978 

5194 

393022 

41 

20 
21 

609734 
8.612823 

5149 

999640 

09 
09 

610094 

5158 
5121 

389906 

40 
39 

5112 

9.999035 

8.613189 

11.386811 

22 

615891 

5076 

999629 

09 

616262 

5085 

383738 

38 

23 

618937 

.5041 

999624 

09 

619313 

5050 

380687 

37 

24 

621962 

5006 

999619 

09 

622343 

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377657 

36 

25 

624965 

4972 

999614 

09 

625352 

4981 

374648 

35 

20 

627948 

4938 

999608 

09 

628340 

4947 

371660 

34 

27 

630911 

4904 

999603 

09 

631308 

49J3 

368692 

33 

28 

633854 

4871 

999597 

09 

634256 

4880 

365744 

32 

29 

636776 

4839 

999592 

09 

637184 

4848 

362816 

31 

30 

639680 

4806 

999586 

09 

640093 

4816 

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30 

31" 

8.642563 

4775 

9.999581 

09 

8.642982 

4784 

11.357018 

29 

32 

645428 

4743 

999575 

09 

645853 

4753 

354147 

28 

33 

648274 

4712 

999570 

09 

648704 

4722 

351296 

27 

34 

651102 

4682 

999564 

09 

651537 

4691 

348463 

26 

35 

653911 

4652 

999558 

10 

654352 

4661 

345648 

25 

36 

656702 

4622 

999553 

10 

657149 

4631 

342851 

24 

37 

659475 

4592 

999547 

10 

65992S 

4602 

340072 

23 

38 

662230 

4563 

999541 

10 

662689 

4573 

337311 

22 

39 

664968 

4535 

999535 

10 

665433 

4544 

334567 

21 

40 
41 

667689 

4506 
4479 

999529 
9.999524 

10 

10 

668160 

4526 

331840 

20 
19 

8.670393 

8.670870 

4488 

11.329130 

42 

673080 

4451 

999518 

10 

673563 

4461 

326437 

18 

43 

675751 

4424 

999512 

10 

676239 

4434 

323761 

17 

44 

678405 

4397 

999506 

10 

678900 

4417 

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16 

45 

681043 

4370 

999500 

10 

681544 

4380 

318456 

15 

46 

683665 

4344 

999493 

10 

684172 

4354 

315828 

14 

47 

686272 

4318 

999487 

10 

686784 

4328 

313216  13 

48 

688863 

4292 

999481 

10 

689381 

4303 

310619'  12 

49 

691438 

4267 

999475 

10 

691963 

4277 

308037 

11 

50 

693998 

4242 

999469 

10 

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4252 

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10 

51 

8  696543 

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9.999463 

11 

8.697081 

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11.302919 

9 

52 

699073 

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11 

099617 

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300383 

8 

53 

701589 

4168 

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11 

702139 

4179 

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7 

54 

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4144 

999443 

11 

704646 

4155 

295354 

6 

55 

706577 

4121 

999437 

11 

707140 

4132 

292860 

5 

56 

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4097 

999431 

11 

709618 

4108 

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4 

57 

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4074 

999424 

11 

712083 

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287917 

3 

58 

713952 

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11 

714534  4062 

285465 

2 

59 

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999411 

11 

716972  4040 

283028 

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60 

718800 

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11 

719396  4017 

280604 

0 

~ 

Cosine  | 

1 

Sine   1 

Cotang.  1       1 

Tang.   |M.| 

SINES   AND   TANGENTS.      3°. 


21 


jrj 

Slno   1 

n.  1 

Cosine  |  D.  | 

Tansr. 

_I^_. 

CofHnir.   1 

0 

8  718800 

4006 

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11 

8.719.396 

40  i  7 

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2 

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11 

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3 

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4 

728337 

3919 

999378 

11 

728959 

3930 

271041 

56 

5 

730688 

3898 

999371 

11 

731317 

3909 

268683 

55 

6 

733027 

3877 

999364 

12 

733663 

3889 

2663.37 

54 

7 

735354 

3857 

999357 

12 

735996 

3868 

264004 

53 

8 

737607 

3836 

999350 

12 

738317 

3848 

261683 

52 

9 

739969 

3816 

999343 

12 

740626 

3827 

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51 

10 

742259 
8.744536 

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9.999329 

12 
12 

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3787 

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50 

49 

J I 

3776 

8.745207 

11.2.54793 

12 

746802 

3756 

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12 

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3768 

2.52521 

48 

13 

749055 

3737 

999315 

12 

749740 

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250260 

47 

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751297 

3717 

999308 

12 

751989 

3729 

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46 

15 

753528 

3698 

999301 

12 

754227 

3710 

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45 

16 

755747 

3679 

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12 

756453 

3692 

243547 

44 

17 

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3661 

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12 

758668 

3673 

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43 

18 

760151 

3642 

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12 

760872 

36.55 

239128 

42 

19 

762337 

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12 

763065 

3636 

236935 

41 

20 

21 

764511 
8.766675 

3606 

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12 
12 

765246 
8.767417 

3618 

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40 
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3588 

9.999257 

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11.232.583 

22 

768828 

3570 

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13 

769578 

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38 

23 

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3553 

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13 

771727 

3565 

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37 

!;4 

773101 

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13 

77.3866 

3548 

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36 

25 

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3518 

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13 

775995 

3.531 

224005 

35 

26 

777333 

3501 

999220 

13 

778114 

3514 

221886 

34 

27 

779434 

3484 

999212 

13 

780222 

3497 

219778 

33 

28 

781524 

3467 

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13 

782320 

3480 

217680 

32 

29 

783605 

3451 

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13 

784408 

3464 

215.592 

31 

30 
31 

785675 
8.787736 

3431 

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13 

786486 

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3431 

213514 
11.211446 

30 
29 

3418 

9.999181 

13 

8.788.5.54 

32 

789787 

3402 

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13 

790613 

3414 

209387 

28 

33 

791828 

3386 

999166 

13 

792662 

3399 

207338 

27 

34 

793859 

3370 

999158 

13 

794701 

3383 

205299 

26 

35 

795881 

3354 

999150 

13 

796731 

3368 

203269 

25 

36 

797894 

3339 

999142 

13 

798752 

3352 

201248 

24 

37 

799897 

3323 

999134 

13 

800763 

3337 

199237 

23 

38 

801892 

3308 

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13 

802765 

3322 

1972.35 

22 

39 

803876 

3293 

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13 

804758 

3307 

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21 

40 

41 

805852 
8.807819 

3278 

999110 

13 
13 

806742 
8.808717 

3292 

1932.58 

20 
19 

3263 

9.999102 

3278 

11.191283 

42 

809777 

3249 

999094 

14 

810683 

3262 

189317 

18 

43 

811726 

3234 

999086 

14 

812641 

3248 

1873.59 

17 

44 

813667 

3219 

999077 

14 

814.589 

3233 

18.5411 

16 

45 

815599 

3205 

999069 

14 

816.529 

3219 

183471 

15 

46 

817522 

3191 

999061 

14 

818461 

3205 

181.539!  14 

47 

819436 

3177 

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14 

820384 

3191 

179616;  13 

48 

821343 

3163 

999044 

14 

822298 

3177 

177702 

12 

49 

823240 

3149 

999036 

14 

824205 

3163 

175795 

11 

50 

825130 

3135 

999027 

14 

826103 

31.50 

17.3897 

10 

51 

8.827011 

3122 

9.999019 

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8.827992 

3136 

11.172008 

9 

52 

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3108 

999010 

14 

829874 

3123 

170126 

8 

53 

830749 

3095 

999002 

14 

831748 

3110 

1682.52 

7 

54 

832607 

3082 

998993 

14 

8.33613 

3096 

166.387 

6 

55 

834456 

3069 

998984 

14 

83.5471 

3083 

164529 

5 

56 

83*^297 

3056 

998976 

14 

837.321 

3070 

162679 

4 

57 

838130 

3043 

9989671 15 

839163 

3057 

160837 

3 

58 

839956 

3030 

998958  15 

840998 

3045 

159002 

2 

59 

841774 

3017 

998950  15 

842825 

3032 

1.57175 

I 

60 

843585 

3000 

998941  15 

844644 

3019 

15.53.56 

0 

Cosine 

1 

1   Sine   1 

Cotang. 

1      1 

Tang.   j  M.  1 

MP, 


22 


4°.      LOGARITHMIC 


M.| 

Sine 

n.  1 

Cosine   1  D.  | 

Tang.   1   D.   1 

(-'"tang.  1  1 

0 

8.843585 

.3005 

9.998941 

15 

8.844644  3019 

11.155356 

60 

1 

845387 

2992 

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15 

846455  3007 

153.545 

59 

2 

847183 

2980 

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15 

848260  2995 

151740 

58 

3 

848971 

2967 

998914 

15 

850057 

2982 

149943 

57 

4 

850751 

2955 

998905 

15 

851846 

2970 

148154 

56 

6 

852525 

2943 

998890 

15 

853628 

2958 

146372 

55 

6 

85429 1 

2931 

998887 

15 

855403 

2946 

144597 

54 

7 

856049 

2919 

998878 

15 

857171 

2935 

142829 

53 

8 

85780 1 

2907 

998869 

15 

858932 

2923 

141068 

52 

9 

859546 

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998860 

15 

860686 

2911 

139314 

51 

10 
11 

861283 
8.863014 

288 1 

998851  15 
9.998841  15 

862433 

2900 

2888 

137.367 
11.1.35827 

50 
49 

2873 

8.864173 

12 

864738 

2861 

998832  15 

865906 

2877 

134094 

18 

13 

866455 

2850 

998823 

16 

867632 

2866 

132368 

47 

14 

868165 

2839 

998813 

16 

869351 

2854 

130649 

46 

15 

869868 

2828 

"  998804 

16 

871064 

2843 

1289.36 

45 

ir. 

871565 

2817 

998795 

16 

872770 

2832 

127230 

44 

17 

873255 

2806 

998785 

16 

874469 

2821 

12.5531 

43 

18 

874938 

2795 

998776 

16 

876162 

2811 

123838 

42 

19 

876615 

2786 

998766 

16 

877849 

2800 

122151 

41 

20 

21 

878285 

2773 
2763 

998757 
9.998747 

16 
16 

879529 

2789 

120471 

40 
39 

8.879949 

8.881202 

2779 

11.118798 

22 

881607 

2752 

998738 

16 

882869 

2768 

117131 

38 

23 

883258 

2742 

998728 

16 

884530 

2758 

115470 

37 

24 

884903 

2731 

998718 

16 

886185 

2747 

113815 

36 

25 

886542 

2721 

998708 

16 

887833 

2737 

112167 

35 

26 

888174 

2711 

998699 

16 

889476 

2727 

110524 

34 

27 

889801 

2700 

998689 

16 

891112 

2717 

108888 

33 

28 

891421 

2690 

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16 

892742 

2707 

107258 

32 

29 

893035 

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17 

894366 

2097 

105634 

31 

30 
31 

894643 
8.896246 

2670 
2660 

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17 
17 

895984 
8.897596 

2687 
2677 

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30 
29 

9.998649 

11.102404 

32 

897842 

2651 

998639 

17 

899203 

2667 

100797 

28 

33 

899432 

2641 

998629 

17 

900803 

2658 

099197 

27 

34 

901017 

2631 

998619 

17 

902398 

2648 

097602 

26 

35 

902596 

2622 

998609 

17 

903987 

2638 

096013 

25 

36 

904169 

2612 

998599 

17 

905570 

2629 

094430 

24 

37 

905736 

2603 

998589 

17 

907147 

2620 

092853 

23 

38 

907297 

2593 

998578 

17 

908719 

2610 

091281 

22 

39 

908853 

2584 

998568 

17 

910285 

2601 

089715 

21 

40 
41 

910404 
8.911949 

2575 

998558 
9.998548 

17 
17 

911846 

2592 

088154 

20 
19 

2566 

8.913401 

2583 

11.086599 

42 

913488 

2556 

998537 

17 

914951 

2574 

085049 

18 

43 

915022 

2547 

998527 

17 

916495 

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083505 

17 

44 

916550 

2538 

998516 

18 

918034 

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16 

45 

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18 

919568 

2.547 

080432 

15 

46 

919591 

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18 

921096 

2538 

078904 

14 

47 

921103 

2512 

998485 

18 

922619 

2530 

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13 

48 

922610 

2503 

998474 

18 

924136 

2521 

075864 

12 

49 

924112 

2494 

998464 

18 

925649 

2512 

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11 

50 

925609 

2486 

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18 

927156 

2.503 

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10 

51 

8.927100 

24-^7 

9.998442 

18 

8.928658 

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11.071342 

9 

bZ 

928587 

2469 

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18 

930155 

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8 

53 

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18 

931647 

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7 

54 

93 1 54 t 

2452 

998410 

18 

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2470 

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6 

55 

933015 

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18 

934616 

2461 

065384 

5 

56 

934481 

2435 

998388 

18 

936093 

2453 

063907 

4 

57 

935942 

2427 

998377 

18 

937565 

2445 

062435 

3 

58 

937398 

2419 

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18 

999032 

2437 

060968 

2 

59 

938850 

2411 

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18 

940494 

2430 

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I 

60 

940296 

2403 

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18 

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2421 

058048 

0 

Coi>i;ie 

1   Sn.e   1 

1  C.tang.  j 

1   Tauii.      }  M. 

86° 


SINES   AND   TANGENTS.      5°. 


23 


E 

«ine 

D. 

Cosine  1  D. 

j   Ta..^. 

1   D. 

1  Cofniig   1 

1  0 

8.940296 

2403 

9 . 998344 

19 

8.941952 

2421 

11. 0580481  60 

1 

94173S 

2394 

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n 

943404 

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056596 

59 

2 

943174 

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19 

944S52 

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055148 

58 

3 

944606 

2379 

998311 

19 

946295 

2397 

053705 

57 

4 

946034 

2371 

998300 

19 

947734 

2390 

052266 

56 

5 

947456 

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19 

949168 

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6 

948874 

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19 

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54 

7 

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19 

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53 

8 

951696 

2340 

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19 

953441 

2360 

046559 

52 

9 

953100 

2332 

993243 

19 

954856 

2351 

045144 

51 

10 

954499 

2325 

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19 

956267 

2344 

043733 

50 

11 

8.955894 

2317 

9.998220 

19 

8.957674 

2337 

11.042326 

49 

12 

9572S4 

2310 

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19 

959075 

2329 

040925 

48 

13 

958670 

2302 

998197 

19 

960473 

2323 

039527 

47 

U 

960052 

2295 

998186 

19 

961866 

2314 

038134 

46 

15 

961429 

2288 

993174 

19 

963255 

2307 

036745 

45 

16 

962801 

2280 

993163 

19 

964639 

2300 

035361 

44 

17 

964170 

2273 

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19 

966019 

2293 

033981 

43 

18 

965534 

2266 

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20 

967394 

2286 

032606 

42 

I'l 

966893 

2259 

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20 

968766 

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41 

20 
21 

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8.969600 

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20 
20 

970133 
8  971496 

2271 

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40 
39 

2244 

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11.028504 

22 

970947 

2238 

998092 

20 

972855 

2257 

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38 

23 

972289 

2231 

998080 

20 

974209 

2251 

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37 

24 

973628 

2224 

993068 

20 

975560 

2244 

024440 

36 

25 

974962 

2217 

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20 

976906 

2237 

023094 

35 

26 

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2210 

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20 

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2230 

021752 

34 

27 

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20 

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2223 

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33 

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20 

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019079 

32 

29 

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2190 

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20 

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31 

30 

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20 

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31 

8.982883 

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9.997934 

20 

8.984899 

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11.015101 

29 

32 

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2170 

997972 

20 

986217 

2191 

013783 

28 

33 

935491 

2163 

997959 

20 

987532 

2184 

012468 

27 

31 

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2157 

997947 

20 

988842 

2178 

011158 

26 

35 

9S8083 

2150 

997935 

21 

990149 

2171 

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25 

30 

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2144 

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21 

991451 

2165 

008549 

24 

37 

990660 

2138 

997910 

21 

992750 

2158 

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23 

38 

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2131 

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21 

994045 

2152 

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22 

39 

993222 

2125 

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21 

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2146 

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21 

40 

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21 

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20 

il 

8  995768 

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9.997860 

21 

8.997908 

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11.002092 

19 

43 

997036 

2106 

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21 

999188 

2127 

000812 

18 

43 

998299 

2100 

997835 

21 

9.000465 

2121 

10.999535 

17 

41 

9995ft0 

2094 

997322 

21 

001738 

2115 

998262 

16 

45 

J  0003.6 

2087 

997809 

21 

003007 

2109 

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15 

46 

002069 

2082 

997797 

21 

004272 

2103 

995728 

14 

47 

003318 

2076 

997784 

21 

005534 

2097 

994466 

13 

IS 

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2070 

997771 

21 

006792 

2091 

993208 

12 

19 

005805 

2064 

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21 

008047 

2085 

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H 

50 

007044 

2058 

997745 

21 

009298 

2080 

990702 

m 

51 

9.003278 

2052 

9.997732 

21 

9.010546 

20T4 

10.939454 

9 

52 

009510 

2046 

997719 

21 

011790 

2068 

938210 

8 

53 

010737 

2040 

997706 

21 

013031 

2062 

986969 

7 

54 

011962 

2034 

997693 

22 

014268 

2056 

985732 

6 

55 

013182 

2029 

997680 

22 

015502 

2051 

984498 

5 

56 

014400 

2023 

9976C7 

22 

0J6732 

2045 

983268 

4 

57 

0  5613 

2017 

997654 

22 

017959 

2040 

982041 

3 

58 

016824 

2012 

997641 

22 

019183 

2033 

980817 

2 

59 

018031 

2006 

997628 

22 

020403 

2028 

979597 

1 

60 

019235 

2030 

997614 

22 

021620 

2023 

978380 

0 

iH 

CDs'iiie  1 

1 

SiMe   1   1 

Cotang. 

1 

Tang.   1  M.  ) 

MP 


24 


6°.      LOGAUITHMIC 


^ 

Brno        1 

D. 

Cosine   |  D. 

Tang   1 

D.   1 

Cntang.   1   ] 

M 

9.019235 

2U00 

9.997614 

22 

9.021620 

2023 

10.978380 

60 

1 

020435 

1995 

997601 

22 

022834 

2017 

977166 

59 

2 

021632 

1989 

997588 

22 

024044 

2011 

975956 

58 

3 

022825 

1984 

997574 

22 

02.5251 

2006 

974749 

57 

4 

024016 

1978 

997501 

22 

026455 

2000 

973.545 

56 

6 

025203 

1973 

997547 

22 

027655 

1995 

972345 

55 

6 

026386 

1967 

997534 

23 

028852 

1990 

971148 

54 

7 

027567 

1962 

997520 

23 

030046 

1985 

9699.54 

53 

8 

028744 

1957 

997.507 

23 

031237 

1979 

968763 

52 

9 

029918 

1951 

997493 

23 

032425 

1974 

967575 

51 

10 
11 

031089, 

1947 

997480 

23 
23 

033609 
9.034791 

1969 
1964 

966391 

50 
49 

9.0322o7{ 

1941 

9.997466 

10.965209 

12 

0334211 

1936 

997452 

23 

035969 

1958 

964031 

48 

13 

034582 

1930 

997139 

23 

0.37144 

19.53 

9628.56 

47 

14 

035741 

1925 

997425 

23 

038316 

1948 

961684 

46 

15 

036896 

1920 

997411 

23 

039485 

1943 

960515 

45 

16 

038048 

1915 

997397 

23 

040651 

1938 

959349 

44 

17 

039197 

1910 

997383 

23 

041813 

19.33 

9.58187 

43 

18 

040342 

1905 

997369 

23 

042973 

1928 

957027 

42 

19 

0414S5 

1899 

997355 

23 

044130 

1923 

95.5870 

41 

20 

21 

04262,") 
9.043762 

1894 

997341 
9.997327 

23 

24 

045284 
9.0464.34 

1918 
1913 

9.54716 
10.9.53.566 

40 
39 

1889 

22 

044895 

1884 

997313 

24 

047582 

1908 

9.52418 

38 

23 

046026 

1879 

997299 

24 

048727 

1903 

951273 

37 

24 

047154 

1875 

997285 

24 

049869 

1898 

950131 

36 

25 

048279 

1870 

997271 

24 

051008 

1893 

948992 

35 

26 

049400 

1865 

997257 

24 

0.52144 

1889 

947856 

34 

27 

050519 

1860 

997242 

24 

053277 

1884 

946723 

33 

28 

051635 

1855 

997228 

24 

054407 

1879 

94.5593 

32 

29 

052749 

1850 

997214 

24 

055535 

1874 

944465 

31 

30 
31 

053859 
054966 

1845 

997199 
9.997185 

24 
24 

056659 
9.0.57781 

1870 

94334 1 

30 
29 

1841 

1865 

10.942219 

32 

056071 

1836 

997170 

24 

058900 

1869 

941100 

28 

33 

057172 

1831 

997156 

24 

060016 

1855 

939984 

27 

34 

058271 

1827 

997141 

24 

061130 

1851 

938870 

26 

35 

059307 

1822 

997127 

24 

062240 

1846 

937760 

26 

36 

060460 

1817 

997112 

24 

063348 

1842 

936652 

24 

37 

061561 

1813 

997098 

24 

064453 

1837 

935547 

23 

38 

062639 

1808 

997083 

25 

06.5556 

1833 

934444 

22 

39 

063724 

1804 

997068 

25 

066655 

1828 

933345 

21 

40 
41 

064806 

1799 

997053 

25 

25 

067752 

1824 
1819 

93^>248 

20 
19 

9.065885 

1794 

9.997039 

9.068846 

10.931154 

42 

066962 

1790 

997024 

25 

069938 

1815 

930062 

18 

43 

068036 

1786 

997009 

25 

071027 

1810 

928973 

17 

44 

069107 

1781 

996994 

25 

072113 

1806 

927887 

16 

45 

070176 

1777 

996979 

25 

073197 

1802 

926803 

15 

46 

071242 

1772 

996964 

25 

074278 

1797 

925722 

14 

47 

072.306 

1768 

996949 

25 

075356 

1793 

924644 

13 

48 

073366 

1763 

996934 

25 

076432 

1789 

923568 

12 

49 

074424 

1759 

996919 

25 

077505 

1784 

922495 

11 

50 
51 

075480 
9.076533 

1755 

996904 
9.996889 

25 
25 

078576 

1780 
1776 

921424 

10 
9 

1750 

9.079644 

10.9203,56 

52 

077583 

1746 

996874 

25 

080710 

1772 

919290 

8 

53 

078631 

1742 

996858 

25 

081773 

1767 

918227 

7 

64 

079676 

1738 

996843 

25 

082833 

1763 

917167 

6 

55 

080719 

1733 

996828 

25 

1   083891 

17.59 

916109 

5 

56 

081759 

1729 

996812 

26 

1   084947 

1  1755 

91.5053 

4 

57 

082797 

1725 

996797 

26 

08600( 

1751 

914000 

3 

58 

083832 

1721 

996782 

26 

087050 

1  1747 

9129.50 

2 

59 

084864 

1717 

996766 

26 

088098 

1743 

911902 

60 

085894 

1713 

.996751 

26 

089144 

1738 

910856 

0 

._ 

Cosine  | 

Bine   1 

j   Co  arig. 

1 

1   Tang.   j  M.  ! 

SINES  AND  TANGENTS. 

f70 

25 

M. 

Sii.^   1 

n.  1 

fWine   1  n.  , 

Tanp.   1 

D.   1 

Coiaiip.   j   I 

0 

9.085894 

1713 

9.996751 

26 

9.089144 

1738 

10.910856 

60 

1 

086922 

1709 

996735 

26 

090187 

1734 

909813 

59 

2 

087947 

1704 

996720 

26 

091228 

1730 

908772 

58 

3 

088970 

1700 

996704 

26 

092266 

1727 

907734 

57 

4 

089090 

1696 

996688 

26 

093.302 

1722 

906698 

56 

5 

091008 

1692 

996673 

26 

094336 

1719 

905664  55 ( 

6 

092024 

1688 

996657 

26 

09.5367 

1715 

904633  54 

7 

093037 

1684 

996641 

26 

096395 

1711 

903605  .531 

8 

094047 

1680 

996625 

26 

097422 

1707 

902578 

52 

9 

095056 

1676 

996610 

26 

098446 

1703 

901.5.54 

51 

10 

006062 

1673 

996594 

26 

099468 

1699 

900532 

50 

11 

9.097065 

1668 

9.996578 

27 

9.100487 

1695 

10.899513 

49 

12 

098066 

1665 

996562 

27 

101504 

1691 

898496 

48 

13 

099065 

1661 

996.546 

27 

102519 

1687 

897481 

47 

14 

100062 

1657 

996530 

27 

103.532 

1684 

896468 

46 

15 

101056 

16.53 

996514 

27 

104.542 

1680 

89.5458 

45 

16 

102048 

1649 

996498 

27 

10.5.5.50 

1676 

894450 

44 

17 

103037 

1645 

996482 

27 

106556 

1672 

893444 

43 

18 

104025 

1641 

996465 

27 

107559 

1669 

892441 

42 

19 

105010 

16.38 

996449 

27 

108560 

1665 

891440 

41 

20 

105992 

1634 

996433 

27 

1095.59 

1661 

890441 

40 

21 

9.106973 

1630 

9.996417 

27 

9.110.5.56 

16.58 

10.889444 

39 

22 

107951 

1627 

996400 

27 

111.551 

1654 

888449 

38 

23 

108927 

1623 

996384 

27 

112,543 

16.50 

887457 

37 

24 

109901 

1619 

996368 

27 

11.3533 

1646 

886467 

36 

25 

1.0873 

1616 

996351 

27 

114.521 

1643 

88.5479 

35 

26 

lil842 

1612 

996335 

27 

115.507 

1639 

884493 

34 

27 

112809 

1608 

996318 

27 

116491 

1636 

883.509 

33 

28 

113774 

1605 

996302 

28 

117472 

1632 

882528 

32 

29 

114737 

1601 

996285 

28 

118452 

1629 

881.548 

31 

30 

1 1.5698 

1.597 

996269 

28 

119429 

1625 

880571 

30 

31 

9.116656 

1.594 

9.996252 

28 

9.120404 

1622 

10.879596 

29 

32 

117613 

1.590 

9962.35 

28 

121377 

1618 

878623 

28 

33 

118567 

1.587 

996219 

28 

122348 

1615 

877652 

27 

34 

119519 

1583 

996202 

28 

123317 

1611 

876683 

26 

35 

120469 

1.580 

996185 

28 

124284 

1607 

875716 

25 

36 

121417 

1.576 

996168 

28 

125249 

1604 

874751 

24 

37 

122362 

1573 

996151 

28 

126211 

1^1 

873789 

23 

38 

123306 

1569 

996134 

28 

127172 

1.597 

872828 

22 

39 

124248 

1.566 

996117 

28 

1281.30 

1.594 

871870 

21 

40 
41 

125187 

1562 
1559 

996100 
9.996083 

28 
29 

129087 
9.130041 

1.591 

870913 

20 
19 

9.126125 

1587 

10.8699.59 

42 

127060 

1.556 

996066 

29 

130994 

1584 

869006 

18 

43 

127993 

1552 

996049 

29 

131944 

1.581 

S68056 

17 

44 

128925 

1.549 

9960.32 

29 

132893 

1577 

867107 

16 

45 

129854 

1545 

996015 

29 

133839 

1.574 

866161 

15 

46 

130781 

1542 

99.5998 

29 

1.34784 

1571 

86.5216 

14 

47 

131706 

1539 

995980 

29 

135726 

1.567 

864274 

13 

4S 

132630 

1,535 

99.5963 

29 

136667 

1564 

863333 

12 

49 

133551 

1.532 

995946 

29 

•  137605 

1561 

862395 

11 

60 

134470 

1529 

995928 

29 

138.542 

1.5.58 

861458 

12 

f>T 

9.135387 

1525 

9.995911 

29 

9.139476 

1.555 

10.860.524 

9 

52 

136303 

1522 

99.5894 

29 

140409 

1551 

8.59.591 

8 

53 

137216 

1519 

99.5876 

29 

141340 

1548 

858660 

7 

54 

138128 

1516 

995859 

29 

142269 

1545 

8.57731 

6 

55 

1390.37 

1512 

99.5841 

29 

143196 

1.542 

856804 

6 

56 

139944 

1509 

99.5823 

29 

144121 

1539 

855879 

4 

57 

140850 

1.506 

995806 

29 

14.5044 

1535 

8.54956 

3 

58 

141754 

1.503 

995788 

29 

145966 

1532 

854034 

2 

59 

142655 

1.500 

99.5771 

29 

146885 

1529 

853115 

1 

60 

1435.55 

1496 

995753 

29 

147803 

1.526 

8.52197 

0 

1  <%.si..e  j 

siiic   j 

1  Louina. 

Tiuig   (RTJ 

26 


LOGARITHMIC 


M 

Sine 

1   D. 

Cosine   1  D 

Tang. 

1   D. 

Coiang. 

n 

T 

9.143555 

1496 

9.995753 

30 

9.147803 

1.526 

10.8.521971  00 

1 

144453 

1493 

995735 

30 

148718 

1523 

851282  59 

2 

145349 

1490 

995717 

30 

149632 

1.520 

8.50368 

68 

3 

140243 

1487 

995699 

30 

150.544 

1517 

849456 

57 

4 

147136 

1484 

995681 

30 

151454 

1514 

848.546 

56 

5 

148026 

1481 

995664 

30 

152363 

1511 

847637 

55 

G 

148915 

1478 

995646 

30 

153269 

1.508 

846731 

54 

T 

149802 

1475 

995628 

30 

154174 

1.505 

845826 

53 

8 

150H86 

1472 

995610 

30 

15.5077 

1.502 

844923 

52 

9 

151569 

1469 

995591 

30 

15.5978 

1499 

844022 

21 

10 

152451 

1466 

995573 

30 

1.56877 

1496 

843123 

50 

11 

9  153330 

1463 

9.995555 

30 

9.157775 

1493 

10.842225 

49 

12 

154208 

1460 

995537 

30 

1.58671 

1490 

841329 

48 

13 

155083 

1457 

995519 

30 

159.565 

1487 

840435 

47 

14 

155957 

1454 

995501 

31 

160457 

1484 

839.543 

46 

15 

156830 

1451 

995482 

31 

161347 

1481 

838653 

45 

16 

157700 

1448 

995464 

31 

162236 

1479 

837764 

44 

17 

158569 

1445 

995446 

31 

163123 

1476 

836877 

43 

18 

159435 

1442 

995427 

31 

164008 

1473 

835992 

42 

19 

160301 

1439 

995409 

31 

164892 

1470 

835108 

41 

20 

161164 

1436 

995390 

31 

165774 

1467 

834226 

40 

21 

9.162025 

1433 

9.995372 

31 

9.166654 

1464 

10.833346 

.39 

22 

162885 

1430 

995353 

31 

167532 

1461 

832468 

38 

23 

163743 

1427 

995334 

31 

168409 

1458 

831591 

37 

24 

164600 

1424 

995316 

31 

169284 

14.55 

830716 

36 

25 

165454 

1422 

995297 

31 

1701.57 

1453 

829843 

35 

26 

166307 

1419 

995278 

31 

171029 

1450 

828971 

34 

27 

167159 

1416 

995260 

31 

171899 

1447 

828101 

33 

28 

168008 

1413 

995241 

32 

172767 

1144 

827233 

32 

29 

168850 

1410 

995222 

32 

173634 

1442 

826366 

31 

30 
31 

169702 
9.170547 

1407 

995203 

32 
32 

174499 

1439 

82.5501 

30 
29 

1405 

9.995184 

9.17.5362 

1436 

10.824638 

32 

171389 

1402 

995165 

32 

176224 

1433 

823776 

28 

33 

172230 

1399 

995146 

32 

177084 

1431 

822916 

27 

34 

173070 

1396 

995127 

32 

177942 

1428 

822058 

26 

35 

173908 

1394 

995108 

32 

178799 

1425 

821201 

25 

36 

174744 

1391 

995089 

32 

1796.55 

1423 

820.345 

24 

37 

17557S 

1388 

99.5070 

32 

180508 

1420 

819492 

23 

38 

176411 

1386 

99.5051 

32 

181360 

1417 

818640 

22 

39 

177242 

1383 

99.5032 

32 

182211 

1415 

817789 

21 

40 
41 

178072 

1380 
1377 

995013 

32 
32 

183059 

1412 

816941 

20 
19 

9.178900 

9.994993 

9.183907 

1409 

10.810093 

42 

179726 

1374 

994974 

32 

1847.52 

1407 

815248 

18 

43 

180551 

1372 

994955 

32 

18.5.597 

1404 

814403 

17 

44 

181374 

1369 

994935 

32 

186439 

1402 

813.561 

16 

45 

182196 

1366 

994916 

33 

187280 

1399 

812720 

15 

46 

183016 

1364 

994896 

33 

188120 

1396 

811880 

14 

47 

183834 

1361 

994877 

33 

1889.58 

1393 

811042!  ni 

48 

184651 

1359 

994857 

33 

189794 

1391 

810206 

U 

49 

185466 

1356 

99483^ 

33 

190629 

1389 

809371 

50 

186280 

1353 

904818 

33 

191462 

1386 

808538 

10 

51 

9.187092 

1351 

9.994798 

33 

9.192294 

1.384 

10.807706 

9 

52 

187903 

1348 

994779 

33 

193124 

1381 

80C876 

8 

53 

188712 

1346 

994759 

33 

193953 

1379 

806047 

7 

54 

189519 

1343 

994739 

33 

194780 

1376 

805220 

6 

65 

190325 

1341 

994719 

33 

195606 

1374 

804394 

5 

66 

191130 

1338 

994700 

33 

196430 

1371 

803570 

4 

67 

191933 

1336 

994680 

33 

1972.53 

1369 

802747 

3 

58 

192734 

1333 

994660 

33 

198074 

1366 

801926 

2 

59 

193534 

1330 

994640 

33 

198894 

1364 

801106 

7 

60 

194332 

1328 

994620 

33 

199713 

1361 

800287 

0 

~1 

Cosine 

i 

Sine    1 

Couna. 

Tan,. 

L_^L 

81° 


SINES  AND   TANGENTS.      9^ 


27 


M. 

gi"«   1 

D.   1 

Cosine   |  D. 

Tanpr. 

d' 

Cotang.   j   1 

0 

9.194332 

1328 

9.994620 

33 

9.199713 

1361 

10. 8002871  60 1 

1 

195129 

J  326 

994G00 

33 

200529 

1359 

7994V 1 

59 

2 

195925 

1323 

994.580 

33 

201345 

1356 

798655 

58 

3 

196719 

1321 

994560 

34 

202159 

13,54 

797841 

57 

4 

197511 

1318 

994540 

34 

202971 

1352 

797029 

56 

5 

198.302 

1316 

994519 

34 

203782 

1349 

796218 

55 

6 

199091 

1313 

994499 

34 

204592 

1347 

795408 

54 

7 

199879 

1311 

994479 

34 

205400 

1345 

794600 

53 

8 

200666 

1308 

994459 

34 

206207 

1342 

793793 

52 

9 

201451 

1306 

994438 

34 

207013 

1340 

792987 

51 

IC 

202234 

1304 
1301 

994418 

34 
34 

207817 

1338 
1335 

792183 

50 
49 

9.203017 

9.994397 

9.208619 

10.791381 

12 

.  203797 

1299 

994377 

34 

209420 

1333 

790580 

48 

13 

204577 

1296 

994357 

34 

210220 

1331 

789780 

47 

14 

205364 

1294 

994336 

34 

211018 

1.328 

788982 

46 

15 

206131 

1292 

994316 

34 

211815 

1326 

788185 

45 

16 

206906 

1289 

994295 

34 

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SINES   AND   TANGENTS.      13°. 


31 


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1   D. 

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1    iang.  jAI. 

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SINES  AND   TANGENTS.      15' 


33 


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1   Cosine   1  D. 

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1   D. 

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1 

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1 

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1 

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34 


16°.      LOGARITHMIC 


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Si.ir 

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Tnnp 

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9.440338 

734 

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38 

23 

450345 

716 

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62 

468347 

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531653 

37 

24 

450775 

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981901 

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468814 

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631186 

36 

25 

451204 

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981924 

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469280 

776 

530720 

35 

26 

451632 

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981886 

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469746 

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27 

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28 

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29 

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30 
3f 

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9.981699 

62 
63 

471605 

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528395 
10.627932 

30 
29 

9.453768 

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9.472068 

772 

32 

454194 

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981662 

63 

472532 

771 

627468 

28 

33 

4rvi6l9 

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981625 

63 

472995 

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627005 

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4:-5044 

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40 
41 

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9.981323 

63 
63 

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9.476683 

766 
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20 
19 

9.458006 

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0 

• 

Cosine 

Sine   ( 

Colang 

'    Tang,   j  M 

73° 


SINES  AND   TANGENTS.      17°. 


35 


■s- 

Slew   1 

n 

Cosine  |  D. 

1   Tang. 

D. 

Coiang.   1   1 

0 

9.465935 

688 

9.980596 

64 

9.4853.39 

755 

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I 

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7.50 

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6 

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"" 

Cociiiie 

1 

Sine   1 

Cotaiig. 

i    '1  arij:   :  W. 

72» 


36 


18' 


LOGARITHMIC 


_M.| 

Sine 

D. 

CVysine   1  I). 

Tang. 

D 

Cotanff.      ] 

0 

9.489982 

648 

9.978206  68 

9.511776 

716 

10.488224 

60 

I 

490371 

648 

978165 

68 

512206 

716 

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2 

490759 

647 

978124 

68 

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715 

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3 

491147 

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513064 

714 

480936 

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4 

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714 

480507 

56 

5 

491922 

645 

978001 

69 

513921 

713 

486079 

55 

6 

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644 

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69 

514349 

713 

485651 

54 

7 

492695 

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69 

514777 

712 

485223 

53 

8 

493081 

643 

977877 

69 

51.5204 

712 

484796 

52 

9 

493466 

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977835 

69 

51.5631 

711 

484369 

51 

10 

493851 
9  494236 

042 

977794 

69 
69 

516057 
9.516484 

710 

483943 
10.48:^516 

50 
49 

|11 

641 

9.9777.52 

7i0 

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494621 

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516910 

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483090  48 

13 

495005 

640 

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517:3:15 

709 

482665  47 

14 

495388 

639 

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639 

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481815 

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496154 

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9.977335 

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22 

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521151 

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24 

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1 

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0 

■^ 

Cosine 

Sine   ( 

[  Co-ang. 

1 

1    ''^-«-    i*M 

71° 


SINES  AND   TANGENTS.      19''. 


37 


"jL 

mne 

D. 

Cosine   |  D. 

Tang. 

"~irn 

Coiariff. 

n 

"o" 

9  512642 

612 

9.975670 

73 

9.536972 

684 

10.463028 

60 

1 

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6il 

97.5627 

73 

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3 

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4 

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638611 

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5 

514472 

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681 

460980 

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6 

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681 

460.571 

54 

7 

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8 

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4.59755 

52 

9 

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H) 
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73 

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U 

n 

Cosiiiti 

1 

Sine    1 

Coiang. 

i 

j    Tanc.    I  M."| 

TOP 


38 


20°.      LOGARITHMIC 


M. 

Sine 

1   D. 

1  (.^osiiie   1  D. 

1   Tang. 

1   D. 

1  ^"*""?-.  1 1 

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0.»iiie 

1   Sine    1 

Coiaijg. 

1   Tan,   -  ^^j 

SINES  AND   TANGENTS.      2V 


39 


M. 

Sine  T 

D.   1 

Cosine  1  I).  1 

Tanp.   1 

D.  1 

Cotang.  1   1 

"U" 

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548 

9.970152  81  1 

9.584177 

629 

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1 

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41.5445 

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3 

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584932 

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3 

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628 

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4 

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627 

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413.561 

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7 

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39 

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22 

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40 
41 

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9.968128 

84 
84 

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613 

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20 
19 

9.667587 

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9.599459 

10.400641 

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625 

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51 

9.670751 

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9.967624 

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9.603127 

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Cosine 

1 

1   ^"-   1 

1  Culaiig. 

1    'I'aiig.   1  M.  1 

68° 


40 


22°.      LOGARITHMIC 


M. 

1   Sine 

1   D. 

1  Cfwine   1  D. 

1   Tan^. 

n. 

Coiang.   1   ] 

0 

U. 573575 

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4 

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27 

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30 
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625,388 

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372 1 48 

0 

. 

Cosine  1 

1   Sine   1 

Coti.iig.   1 

1    Tang.   j  M.  j 

67° 


SINES  AND  TANGENTS. 

23°. 

41 

M. 

1    Sine 

I). 

1   Cosine   |  D. 

1   Tana. 

D. 

1   C<.tang.  1   1 

0 

9.591878 

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9.964026 

89 

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5 

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6 

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7 

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9 

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10 

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50 
49 

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14 

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15 

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16 

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633795 

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36.5857 

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36.5510 

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20 
21 

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9.6.35185 

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10.364816 

40 
39 

9.. 598075 

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22 

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9.603882 

480 

961846 

92 
92 

641747 

572 

358253 

20 
19 

480 

9.961791 

9.642091 

572 

10.3.57909 

42 

604170 

479 

9617.35 

92 

642434 

572 

357566 

18 

43 

604457 

479 

961680 

92 

642777 

572 

357223 

17 

44 

604745 

479 

961624 

93 

643120 

571 

356880 

16 

45 

605032 

478 

961.569 

93 

64.3463 

571 

356.537 

15 

46 

605319 

478 

961513 

93 

643806 

671 

3.56194 

14 

47 

605606 

478 

9614.58 

93 

644148 

670 

3558.52 

13 

48 

605892 

477 

961402 

93 

644490 

670 

35.5510 

12 

49 

606179 

477 

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93 

644832 

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3.55168 

11 

50 
51 

606465 

476 

961290 

93 
93 

645174 

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3.54826 
10.3.54484 

10 
9 

9.606751 

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9.961235 

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52 

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3.54143 

8 

53 

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475 

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93 

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54 

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646.540 

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55 

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3.53119 

6 

56 

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647222 

568 

352778 

4 

67 

608461 

474 

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647562 

567 

352438 

3 

58 

608745 

473 

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647903 

567 

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2 

59 

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9.609313 

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9.960730 

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1 

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2 

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3 

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94 

649602 

566 

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57 

4 

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649942 

565 

350058 

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5 

610729 

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565 

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65 

6 

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470 

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565 

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7 

611294 

470 

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94 

650959 

564 

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53 

8 

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470 

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94 

651297 

564 

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9 

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94 

651636 

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10 
11 

612140 

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9.960109 

94 
95 

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563 

348026 
10.347688 

50 
49 

9.612421 

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9.652312 

563 

12 

612702 

468 

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95 

652650 

563 

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13 

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468 

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95 

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563 

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14 

613264 

467 

959938 

95 

653326 

562 

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46 

15 

613545 

467 

959882 

95 

653663 

562 

346837 

45 

Ifi 

613825 

467 

959825 

95 

654000 

562 

346000 

44 

17 

614105 

466 

959768 

95 

654337 

561 

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43 

18 

614385 

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95 

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561 

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19 

614665 

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21 

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25 

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27 

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30 
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41 

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19 

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49 

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50 
51 

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97 
97 

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9.665697 

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10.334303 

10 
9 

9.623502 

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9.957804 

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57 

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98 

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550 

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58 

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650 

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2 

69 

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452 

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98 

668343 

550 

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1 

60 

625948 

451 

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668672 

550 

331328 

u 

1   Cosine 

1   Sine   1 

Cotang. 

1 

1    'lang   1  M."j 

61^ 


SINES  AND  TANGENTS.      25°. 


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Sine 

D. 

Cosine  |  D. 

1   Tang. 

P. 

1   Cotang  1   ' 

0 

9.625948 

451  1 

9.957276 

98 

9.668673 

5.50 

10.331327|60I 

1 

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451 

957217 

98 

669002 

549 

330998 

59/ 

2 

626490 

451  1 

957158 

98 

669332 

549 

3306G8 

58 

3 

626760 

450  1 

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98 

669661 

549 

330339 

57 

4 

627030 

45C 

957040 

98 

669991 

548 

330009 

56 

5 

627300 

450  1 

956981 

98 

670320 

548 

329680 

55 

6 

627570 

449 

956921 

99 

670649 

548 

329351 

54 

7 

627840 

449 

956862 

99 

670977 

548 

329023 

53 

8 

628109 

449 

956803 

99 

671306 

547 

328694 

52 

9 

628378 

448 

956744 

99 

671634 

547 

328366 

51 

10 
11 

628647 
9.628916 

448 
447 

956684 

99 
99 

671963 

547 

328037 

50 
49 

9.956625 

9.672291 

547 

10.327709 

12 

629185 

447 

956566 

99 

672619 

546 

327381 

48 

13 

629453 

447 

956506 

99 

672947 

546 

327053 

47 

14 

629721 

446 

956447 

99 

673274 

646 

326726 

46 

15 

629999 

446 

956387 

99 

673602 

546 

326398 

45 

16 

630257 

446 

956327 

99 

673929 

545 

326071 

44 

17 
18 

tmn 

446 

956268 

99 
100 

674257 
674584 

545 
545 

325743 
325416 

43 

42 

445 

956208 

19 

631059 

445 

956148 

100 

674910 

544 

325090 

41 

20 
21 

631326 
9.631593 

445 

956089 

100 
100 

675237 
9.675564 

544 
544 

324763 

40 
39 

444 

9.956029 

10.324436 

22 

631859 

444 

955969 

100 

675890 

544 

324110 

38 

23 

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444 

955909 

100 

676216 

543 

323784 

37 

24 

632392 

443 

955849 

100 

676543 

543 

323457 

36 

25 

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443 

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100 

676869 

543 

323131 

35 

26 

632923 

443 

955729 

100 

677194 

543 

322806 

34 

27 

633189 

442 

955069 

100 

677520 

542 

322480 

33 

28 

633454 

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955609 

100 

677846 

542 

322154 

32 

29 

633719 

442 

955548 

100 

678171 

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321829 

31 

30 

633984 

441 

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100 

678496 

542 

321504 

30 

31 

9.634249 

441 

9.955428 

101 

9.678821 

641 

10. .321 179 

29 

32 

634514 

440 

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lOl 

679146 

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28 

33 

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101 

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34 

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101 

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26 

35 

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439 

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101 

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540 

319880 

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101 

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680768 

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38 

636097 

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101 

681092 

540 

318908 

22 

39 

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101 

681416 

539 

318584 

21 

40 
41 

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438 

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9  954823 

101 
101 

681740 

639 
539 

318260 

20 
19 

9.638886 

437 

9.682063 

10.317937 

42 

637148 

437 

954762 

101 

682387 

539 

317613 

18 

43 

63741 1 

437 

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101 

682710 

538 

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44 

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101 

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538 

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16 

15 

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538 

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537 

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48 

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102 

684324 

537 

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102 

684046 

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435 

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102 

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51 

9.639503 

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9.685290 

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10.3147101  9| 

52 

639764 

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102 

685612 

536 

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53 

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434 

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535 

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4 

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534 

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1 

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534 

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D 

j   Cotaiig. 

r 

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9.641842 
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431 

9.9.53660 

103 

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534 

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533 

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4 

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533 

310.537 

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5 

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633 

310217 

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6 

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633 

309897 

64 

7 

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103 

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533 

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8 

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9.53166 

103 

690742 

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3092.58 

52 

9 

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9,53104 

103 

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532 

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10 
11 

644423 

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9.9.52980 

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104 

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49 

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SINES  AND  TANGENTS.      27 


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49 

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9.949170 

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12 

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108 

710904 

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289096 

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13 

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409 

949040 

108 

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660501 

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108 

711.525 

517 

288475 

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15 

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108 

711836 

617 

288164 

45 

16 

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408 

948845 

108 

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517 

2878.54 

44 

17 

661236 

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109 

7124.56 

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126 

778487 

477 

221512 

1 

60 

711839 

1  350 

933066 

126 

778774 

477 

221226 

0 

1   Cosine 

1   Si..e   1 

Cotana. 

1 

j   Taiig.   1  M. 

69» 


SINES   AND   TANGENTS.      31' 


49 


M." 

Sine 

Q- 

(V)sine   |  D. 

Tang. 

_2^ 

Cotans.   1   1 

"T 

9  711839 

350 

9.933066 

126 

9.778774 

477 

10.221226 

60 

1 

712050 

350 

932990 

127 

779060 

477 

220940 

59 

2 

712260 

350 

932914 

127 

779346 

176 

220654 

58 

3 

712469 

349 

932838 

127 

779632 

476 

220368 

57 

4 

712679 

349 

932762 

127 

779918 

476 

220082 

56 

5 

712889 

349 

932685 

127 

780203 

476 

219797 

55 

6 

713098 

349 

932609 

127 

780489 

476 

219511 

54 

7 

713308 

349 

932533 

127 

780775 

476 

219225 

53 

8 

713517 

348 

932457 

127 

781060 

476 

218940 

52 

9 

713726 

348 

932380 

127 

781346 

475 

2186.54 

51 

10 

713935 

348 

932304 

127 

781631 

475 

218369 

50 

11 

9.714144 

348 

9.932228 

127 

9.781916 

475 

10.218084 

49 

12 

714352 

.347 

932151 

127 

782201 

475 

217799 

48 

13 

714561 

347 

932075 

128 

782486 

475 

217514 

47 

14 

714769 

347 

931998 

128 

782771 

475 

217229 

46 

15 

714978 

347 

931921 

128 

783056 

475 

216944 

45 

16 

715186 

347 

931845 

128 

783341 

475 

216659 

44 

17 

715394 

346 

931768 

128 

783626 

474 

216374 

43 

18 

715602 

346 

931691 

128 

783910 

474 

216090 

42 

19 

715809 

346 

931614 

128 

784195 

474 

215805 

41 

20 
21 

716017 
9.716224 

346 

931537 

128 
128 

784479 

474 

215521 

40 
39 

345 

9.931460 

9.784764 

474 

10.2152.36 

22 

716432 

345 

931383 

128 

785048 

474 

214952 

38 

23 

716639 

345 

931306 

128 

785332 

473 

214668 

37 

24 

716846 

345 

931229 

129 

785616 

473 

214384 

36 

20 

717053 

345 

931152 

129 

785900 

473 

214100 

35 

26 

717259 

344 

931075 

129 

786184 

473 

213816 

34 

27 

717466 

344 

930998 

129 

786468 

473 

213532 

33 

28 

717673 

344 

930921 

129 

786752 

473 

.   213248 

32 

29 

717879 

344 

930843 

129 

787036 

473 

212964 

31 

30 

31 

718085 

343 

930766 

129 
129 

787319 

472 

212681 

30 
29 

9.718291 

343 

9.930688 

9.787603 

472 

10.212397 

32 

718497 

343 

930611 

129 

787886 

472 

212114 

28 

33 

718703 

343 

930533 

129 

788170 

472 

211830 

27 

34 

718909 

343 

930456 

129 

788453 

472 

211547 

26 

35 

719114 

342 

930378 

129 

788736 

472 

211264 

25 

36 

719320 

342 

930300 

130 

789019 

472 

210981 

24 

37 

719525 

342 

930223 

130 

789302 

471 

210698 

23 

38 

719730 

342 

930145 

130 

789585 

471 

210415 

22 

39 

719935 

341 

930067 

130 

789868 

471 

210132 

21 

40 
41 

720140 
9.720345 

341 
341 

929989 
9.929911 

130 
130 

790151 

471 

209849 

20' 
19 

9.790433 

471 

10.209567 

42 

720549 

341 

929833 

130 

790716 

471 

209284 

18 

43 

720754 

340 

929755 

130 

790999 

471 

209001 

17 

44 

720958 

340 

929677 

130 

791281 

471 

208719 

16 

45 

721162 

340 

929599 

130 

791563 

470 

208437 

15 

46 

721366 

340 

929521 

130 

791846 

470 

208154 

14 

47 

721570 

340 

929442 

130 

792128 

470 

207872 

13 

48 

721774 

339 

929364 

131 

792410 

470 

207590 

12 

49 

721978 

339 

929286 

131 

792692 

470 

207308 

11 

50 
51 

722181 
9.722385 

339 

929207 

131 
131 

792974 

470 

207026 

10 
9 

339 

9.929129 

9.793256 

470 

10.206744 

52 

722588 

339 

929050 

131 

793538 

469 

206462 

8 

53 

722791 

338 

928972 

131 

793819 

469 

206181'  7 

54 

722994 

338 

928893 

131 

794I0I 

469 

205899  6 

55 

723197 

338 

928815 

131 

794383 

469 

205617 

5 

56 

723400 

338 

928736 

131 

794664 

469 

205336 

4 

57 

723603 

337 

928657 

131 

794945 

469 

20.5055 

3 

58 

723805 

337 

928578 

131 

795227 

469 

204773 

2 

59 

724007 

337 

928499 

131 

795508 

468 

204492 

1 

60 

724210 

337 

928420 

131 

795789 

468 

204211 

0 

1 

Citeiiic  1 

Sine   1 

Coiang. 

Thus. 

2! 

68° 


50 


32°.      LOGARITHMIC 


M. 

[   Sine 

D. 

Cosine   |  D. 

1   Tanjr. 

1  j>. 

1   Cotang.  1 

0 

9.724210 

337 

9.928420 

132 

9.795789 

468 

10.204211,60 

1 

724412 

337 

928342 

132 

796070 

468 

203930 

59 

2 

724614 

336 

928263 

132 

796351 

468 

.203649 

58 

3 

724816 

336 

928183 

132 

796632 

468 

20.3368 

57 

4 

725017 

335 

928104 

132 

796913 

468 

203087 

5b 

5 

725219 

336 

928025 

132 

797194 

468 

202806' 55 1 

6 

725420 

335 

927946 

132 

797475 

468 

202525 

54 

7 

725622 

335 

927867 

132 

797755 

468 

202245 

53 

8 

725823 

335 

927787 

132 

798036 

467 

201964 

52 

9 

726024 

335 

927708 

132 

798316 

467 

201684 

51 

10 

II 

726225 

335 

927629 

132 
132 

798596 

467 

20 1404 

50 

49 

9.726426 

334 

9.927549 

9.798877 

467 

.0.201123 

12 

726626 

334 

927470 

133 

799157 

467 

200843 

48 

13 

726827 

334 

927390 

1.33 

799437 

467 

200563 

47 

14 

727027 

334 

927310 

133 

799717 

467 

200283 

46 

15 

727228 

334 

927231 

133 

799997 

466 

200003 

45 

16 

727428 

333 

927151 

133 

800277 

466 

199723 

44 

17 

727628 

333 

927071 

133 

800557 

466 

199443 

43 

18 

727828 

333 

926991 

133 

800836 

466 

199164 

42 

19 

728027 

333 

926911 

133 

801116 

466 

198884 

41 

20 

728227 

333 

926831 

133 

801396 

466 

198604 

40 

21 

9.728427 

332 

9.926751 

133 

9.801675 

466 

10.198325 

39 

22 

728626 

332 

926671 

133 

801955 

466 

198045 

38 

23 

728825 

332 

926591 

133 

802234 

465 

197766 

37 

24 

729024 

332 

926511 

1.34 

802513 

465 

197487 

36 

25 

729223 

331 

926431 

134 

802792 

465 

197208 

35 

26 

729422 

.331 

926351 

134 

803072 

465 

196928 

34 

27 

729621 

331 

926270 

134 

803351 

465 

196649 

33 

28 

729820 

331 

926190 

134 

803630 

465 

196370 

32 

29 

730018 

330 

926110 

134 

803908 

465 

196092 

31 

30 
31 

730216 
9.730415 

330 
330 

926029 
9.925949 

134 
134 

804187 
9.804466 

465 
464 

19.5813 

30 
29 

10.195534 

32 

730613 

330 

925868 

134 

804745 

464 

195255 

28 

33 

730811 

330 

925788 

134 

805023 

464 

194977 

27 

34 

731009 

329 

925707 

134 

805302 

464 

194698 

26 

35 

731206 

329 

925626 

134 

805580 

464 

194420 

25 

36 

731404 

329 

925545 

135 

805859 

464 

194141 

24 

37 

731602 

329 

925465 

135 

806137 

464 

193863 

23 

38 

731799 

329 

925384 

135 

806415 

463 

193585  22 

39 

731996 

328 

925303 

1.35 

806693 

463 

193307"  21 

40 

732193 

328 

925222 

135 

806971 

463 

193029 

20 

41 

9.732390 

328 

9.925141 

135 

9.807249 

463 

10.. 92751 

19 

42 

732587 

328 

925060 

135 

807527 

463 

192473 

18 

43 

732784 

328 

924979 

135 

807805 

463 

192195 

17 

44 

732980 

327 

924897 

135 

808083 

463 

191917 

16 

45 

733177 

327 

924816 

135 

808361 

463 

191639 

15 

46 

733373 
T§§i69 

327 

924735 

136 

808638 

462 

19i362  14 
191084^ 

47 

327 

92^654 

136 

808916 

462 

48 

733765 

327 

924572 

136 

809193 

462 

190807  12 

49 

733961 

326 

924491 

136 

809471 

462 

190529'  11 

50 

734157 

326 

924409 

136 

809748 

462 

190252  K 

51 

9.734353 

326 

9.924328 

136 

9.810025 

462 

10.189975  9 

52 

734549 

326 

924246 

136 

810302 

462 

189698'  8 

53 

734744 

325 

924 ir4 

136 

810580 

462 

1894201  7 

64 

734939 

325 

924083 

136 

810857 

462 

189143 

^ 

55 

735135 

325 

924001 

136 

811134 

461 

188866 

6 

56 

735330 

325 

923919 

136 

811410 

461 

188590 

4 

57 

735525 

325 

923837 

136 

811687 

461 

188313 

3 

58 

735719 

321 

923755 

137 

811964 

461 

188036 

2 

69 

735914 

324 

923673 

137 

812241 

461 

187759 

1 

60 

736109 

324 

92359 1 

137 

812517 

461 

187483 

0 

«= 

Cosine 

Sine   1    1 

Cotang. 

1    Tang.   ;M.J 

SINES   AND  TANGENTS.      33°. 


61 


M. 

Sine 

0. 

Cogine   1  D. 

1   Tang. 

D. 

Cotairg.  1 

0 

9.736109 

324 

9.92.3,591 

137 

9.812517 

461 

10.187482  60 

i 

736303 

324 

923509 

137 

812794 

461 

187206 

59 

2 

73G498 

324 

923427 

137 

81.3070 

461 

186930 

68 

3 

736692 

323 

923.345 

137 

813347 

460 

186653 

57 

4 

736880 

323 

923263 

137 

813623 

460 

186377 

56 

5 

737080 

323 

923.-81 

137 

813899 

460 

186101 

55 

6 

737274 

323 

923098 

137 

814175 

460 

185825 

54 

7 

737467 

323 

923016 

137 

814452 

460 

18.5.548 

53 

8 

737661 

322 

922933 

137 

814728 

460 

18.5272 

52 

9 

737855 

322 

922851 

137 

815004 

460 

184996 

51 

10 
11 

738048 

322 

922768 
9.922686 

138 

138 

815279 

460 

184721 

50 
49 

9.738241 

322 

9.815555 

459 

10.184445 

12 

738434 

322 

922603 

138 

815831 

459 

184169 

48 

13 

738627 

321 

922520 

138 

816107 

459 

183893 

47 

14 

738820 

321 

922438 

1.38 

816382 

459 

183618 

46 

15 

739013 

,321 

922355 

138 

816658 

459 

183342 

45 

16 

739206 

321 

922272 

138 

816933 

459 

183067 

44 

17 

739398 

321 

922189 

138 

817209 

459 

182791 

43 

18 

739590 

.320 

922106 

1.38 

817484 

459 

182516 

42 

19 

739783 

320 

922023 

138 

8177,'59 

459 

182241 

41 

20 

739975 

320 

921940 

138 

818035 

458 

181965 

40 

21 

9.740167 

320 

9.921857 

139 

9.818310 

458 

10.181690 

39 

22 

740359 

320 

921774 

139 

818585 

458 

181415 

38 

23 

740550 

319 

921691 

139 

818860 

458 

181140 

37 

24 

740742 

319 

921607 

1.39 

819135 

458 

180865 

36 

25 

740934 

319 

921.524 

139 

819410 

458 

180.590 

35 

26 

741125 

319 

.921441 

139 

819684 

458 

180316 

34 

27 

741316 

319 

921357 

139 

819959 

458 

180041 

33 

28 

741508 

318 

921274 

139 

820234 

458 

179766 

32 

29 

741699 

318 

921190 

139 

820508 

457 

179492 

31 

30 
31 

741889 

318 

921107 
9.921023 

139 
139 

820783 
9.821057 

457 

179217 

30 
29 

9.742080 

318 

457 

10.178943 

32 

742271 

318 

920939 

140 

8213.32 

457 

178668 

28 

33 

742462 

317 

920856 

140 

821606 

457 

178394 

27 

34 

742652 

317 

920772 

140 

821880 

457 

178120 

26 

35 

742842 

317 

920688 

140 

822154 

457 

177846 

25 

36 

743033 

317 

920604 

140 

822429 

457 

177571 

24 

37 

743223 

317 

920520 

140 

822703 

4.57 

177297 

23 

38 

743413 

316 

920436 

140 

822977 

456 

177023 

22 

39 

743602 

316 

920352 

140 

823250 

456 

176750 

21 

40 

743792 

316 

920268 

140 

823524 

456 

176476 

20 

41 

9.74.3982 

316 

9.930184 

140 

9.823798 

456 

10.176202 

19 

42 

744171 

316 

920099 

140 

824072 

4.56 

175928 

18 

43 

744361 

315 

920015 

140 

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456 

1756.55 

17 

44 

744550 

315 

919931 

141 

824619 

456 

17.5.381 

16 

45 

744739 

315 

919846 

141 

824893 

456 

175107 

15 

46 

744928 

315 

919762 

141 

825166 

456 

174834 

14 

47 

745117 

315 

919677 

141 

825439 

455 

174.561 

13 

4^ 

745306 

314 

919593 

141 

825713 

455 

174287 

12 

49 

74.5494 

314 

919508 

141 

82.5986 

455 

174014 

11 

50 

745683 

314 

919424 

141 

826259 

455 

173741 

10 

51 

9.74.5871 

314 

9.919339 

141 

9.826.532 

455 

IQ. 173468 

9 

52 

746059 

314 

9192.54 

141 

826805 

455 

173195 

8 

53 

746248 

313 

919169 

141 

827078 

4.55 

172922 

7 

54 

746436 

313 

919085 

141 

827351 

455 

172649 

6 

55 

746624 

313 

919000 

141 

827624 

455 

172376 

5 

56 

746812 

313 

918915 

142 

827897 

4.54 

172103 

4 

57 

746999 

313 

9188.30 

142 

828170 

454 

171830 

3 

58 

747187 

312 

918745 

142 

82S442 

454 

171.5.58 

2 

59 

747374 

312 

9186.59 

142 

828715 

454 

171285 

1 

60 

747562 

312 

918.574 

142 

828987 

454 

171013 

0 

~ 

Cosine 

nine        1 

CoVmg. 

1    Tang.   1  M.  | 

52 


34°.      LOGARITHMIC 


M. I       Sine 


D.      I      Cosine     |  D.    |      Tang.      \ 


I         I'otang.    I 


3.' 


306 
.306 
306 
305 
305 
305 
305 
305 
304 
304 
'304 
304 
304 
304 
303 
303 
303 
303 
303 
302 
302 
302 
302 
302 
301 
301 
301 
301 
301 
301 


9.918574 

142 

9.828987 

454 

918489 

142 

829260 

454 

918404 

142 

829532 

454 

918318 

142 

829805 

454 

918233 

142 

830077 

454 

918147 

142 

830349 

453 

918062 

142 

830621 

453 

917976 

143 

830893 

453 

917891 

143 

831165 

453 

917805 

143 

831437 

453 

917719 

143 

831709 

453 

9.917634 

143 

9.831981 

453 

917548 

143 

832253 

453 

917462 

143 

832525 

453 

917376 

143 

832796 

453 

917290 

143 

833068 

452 

917204 

143 

833339 

452 

917118 

144 

8.33611 

452 

917032 

144 

833882 

452 

916946 

144 

834154 

452 

916859 
9.916773 

144 

144 

834425 

452 

9.834696 

452 

916687 

144 

834967 

452 

916600 

144 

835238 

452 

916514 

144 

835.509 

452 

916427 

144 

835780 

451 

916341 

144 

838051 

451 

916254 

144 

836322 

451 

916167 

145 

836593 

451 

916081 

145 

836864 

451 

915994 

145 

8371.34 

451 

9.915907 

145 

9.837405 

451 

915820 

145 

837675 

451 

915733 

145 

837946 

451 

915646 

145 

838216 

451 

915559 

145 

838487 

450 

915472 

145 

838757 

450 

915385 

145 

839027 

450 

915297 

145 

839297 

450 

915210 

145 

839568 

450 

915123 

146 

8398,38 

450 

9.915035 

146 

9.840108 

450 

914948 

146 

840378 

450 

914860 

146 

840647 

450 

914773 

146 

840917 

449 

914685 

146 

841187 

449 

914598 

146 

841457 

449 

914510 

146 

841726 

449 

914422 

146 

841.996 

449 

914334 

146 

842266 

449 

914246 

147 

842535 

449 

9.914158 

147 

9.842805 

449 

914070 

147 

843074 

449 

91.3982 

147 

843343 

449 

913894 

147 

843612 

449 

913806 

147 

843882 

448 

913718 

147 

844151 

448 

91.3630 

147 

844420 

448 

913541 

147 

844689 

448 

9134.53 

147 

844958 

448 

913365 

147 

84.5227 

448 

10.171013 
170740 
170463 
170195 
169923 
169651 
169379 
169107 
168835 
168583 

168291 

10.1f.8019 
167747 
167475 
167204 
166932 
166661 
166389 
166118 
165846 

165575 

10.165304 
165033 
164762 
164491 
164220 
163949 
163678 
163407 
163136 

162886 

107162595 
162325 
162054 
161784 
161513 
161243 
160973 
160703 
160432 
: 60 162 


10. 


Sine 


Cuiang. 


1.59802 
159622 
1.59353 
159083 
1.58813 
1.58.543 
158274 
1.58004 
1.57734 

157465 

10.1.57195 
1.56926 
156657 
1.56388 
1.56118 
15.5849 
15.5580 
155311 
155042 
1547731 

— ^n^.~ 


65° 


SINES  AND   TANGENTS.      35°. 


63 


M. 

Pl!.e   1 

D.   1  Cosine   1  D.  1 

Tane. 

D. 

ColtniB.  ] 

'o~ 

9.Vr)8.'>9J 

301 

9.913365 

147 

9.84.5227 

448 

10  154773|  60 

i 

7.58772 

300 

913276 

147 

84.5496 

448 

1.54.504 

59 

2 

758952 

300 

913187 

148 

845764 

448 

1.54236 

58 

3 

759132 

300 

913099 

148 

8460p 

448 

153967 

57 

4 

759312 

300 

913010 

148 

S463IJ2 

448 

I53G98 

56 

5 

759492 

300 

912922 

148 

846570 

447 

153430 

55 

6 

759672 

299 

912833 

148 

846839 

447 

1.53161 

54 

7 

759852 

299  1   9127441 

148 

847107 

447 

152893 

53 

8 

760031 

299 

9)26.55 

148 

847376 

447 

152624 

52 

9 

760211 

299 

912566 

148 

847644 

447 

152356 

51 

10 

760390 

299 

912477 

148 

847913 

447 

152087 

50 

11 

9.760.569 

298 

3.912388 

148 

9.848181 

447 

10.151819 

49 

12 

760748 

298 

912299 

149 

848449 

447 

151.551 

48 

13 

760927 

298 

912210 

149 

848717 

447 

151283 

47 

14 

761106 

298 

912121 

149 

848986 

447 

151014 

46 

15 

761285 

298 

912031 

149 

849254 

447 

1.50746 

45 

16 

761464 

298 

911942 

149 

849522 

447 

150478 

44 

17 

761642 

297 

911853 

149 

849790 

446 

1.50210 

43 

18 

761821 

297 

911763 

149 

850058 

446 

149942 

42 

19 

761999 

297 

911674 

149 

850325 

446 

149675 

41 

20 
21 

762177 

297 

911584 
9.911495 

149 
149 

8.50593 

446 

149407 

40 
39 

9.762356 

297 

9.8.50861 

446 

10.149139 

22 

762534 

296 

911405 

149 

851129 

446 

148871 

38 

23 

762712 

296 

911315 

150 

851396 

446 

148604 

37 

24 

762889 

296 

911226 

150 

851664 

446 

148336 

36 

2.5 

763067 

296 

911136 

150 

851931 

446 

148069 

35 

26 

763245 

296 

911046 

150 

852199 

446 

147801 

34 

27 

763422 

296 

910956 

1.50 

852466 

446 

147,534 

33 

28 

763600 

295 

910866 

150 

852733 

445 

147267 

32 

29 

763777 

295 

910776 

150 

853001 

445 

146999 

31 

iSO 

763954 

295 

910686 

150 
150 

8.53268 

445 

146732 

30 
29 

IsT 

9.764131 

295 

9.910596 

9.853.535 

445 

10.146465 

32 

764308 

295 

910.506 

150 

853802 

445 

146198 

28 

33 

764485 

294 

910415 

150 

854069 

445 

145931 

27 

34 

764662 

294 

910325 

151 

854336 

445 

14.5681 

26 

35 

764838 

294 

910235 

151 

854603 

445 

145397 

25 

36 

76.5015 

294 

910144 

151 

854870 

445 

145130 

24 

37 

76519] 

294 

9100.54 

151 

855137 

445 

144863 

23 

38 

765367 

294 

909963 

151 

855404 

445 

144.596 

22 

39 

76.5544 

293 

909873 

151 

855671 

444 

144.329 

21 

40 
41 

785720 

293 
293 

909782 
9.909691 

151 
151 

855938 

444 

144062 
10.143796 

20 

19 

9.765896 

9.8.56204 

444 

42 

766072 

203 

909601 

151 

856471 

444 

143529 

18 

43 

766247 

293 

909510 

151 

8.56737 

444 

143263 

17 

44 

766423 

293 

909419 

151 

857004 

444 

1429961  16 

I'i 

766598 

292 

909328 

1.52 

857270 

444 

142730 

15 

46 

766774 

292 

909237 

1.52 

857.537 

444 

142463 

14 

17 

766949 

292 

909146 

1.52 

857803 

444 

142197 

13 

48 

767124 

292 

909055 

1.52 

8.58069 

444 

141931 

12 

49 

767300 

292 

908964 

152 

858336 

444 

141664 

11 

50 
61 

767475 

291 

908873 
9.908781 

1.52 
152 

858602 

443 

141398 

10 
9 

9.767649 

291 

9.858868 

443 

10.141132 

62 

767824 

291 

908690 

152 

8.59134 

443 

140866 

8 

53 

767999 

291 

908.599 

1.52 

859400 

443 

140600 

7 

64 

768173 

291 

908507 

152 

8.59666 

443 

140334 

6 

65 

768348 

290 

908416 

153 

859932 

443 

140068 

5 

56 

768.522 

290 

908324 

153 

860198 

443 

1.39802 

4 

57 

768697 

290 

908233 

1.53 

860464 

443 

139536 

3 

58 

768871 

290 

908141 

1.53 

860730 

443 

139270 

2 

59 

769045 

290 

908049 

153 

860995 

443 

139005 

1 

60 

769219 

290 

907958'  1.53 

861261 

443 

138739 

0 

Cosine 

1   Sine   1 

1  Colung. 

1 

1   'I'ang. 

yr 

64" 


64 


36°.      LOGARITHMIC 


M. 

Sine   1 

D. 

Cosine   |  T). 

i  'lans. 

D 

1  Cotanp.   j   j 

0| 

9.769219 

290 

9.907958 

153 

9.861261 

443 

10.138739 

60 

1 

769393 

289 

907866 

153 

861527 

443 

138473 

59 

2 

769506 

289 

907774 

153 

861792 

442 

138208 

58 

3 

769740 

289 

907682 

1.53 

862058 

442 

137942 

57 

4 

769913 

289 

907590 

153 

862323 

442 

137677 

56 

5 

770087 

289 

907498 

153 

802.589 

442 

137411 

55 

6 

770260 

288 

907400 

153 

862854 

442 

137146 

54 

7 

770433 

288 

907314 

154 

863119 

442 

136881 

53 

8 

770606 

288 

907222 

154 

863385 

442 

136615 

52 

9 

770779 

288 

907129 

154 

8636.50 

442 

136350 

51 

10 

•  770952 

288 

907037 

154 

863915 

442 

136085 

50 

11 

9.771125 

288 

9  906945 

154 

9.864180 

442 

10.13.5820 

49 

12 

771298 

287 

906852 

154 

864445 

442 

13.5555 

48 

13 

771470 

287 

906760 

154 

864710 

442 

13.5290 

47 

14 

771643 

287 

906667 

154 

864975 

441 

13.5025 

46 

15 

771815 

287 

906575 

154 

865240 

441 

134760 

45 

16 

771987 

287 

906482 

154 

865505 

441 

134495 

44 

17 

772159 

287 

906389 

155 

865770 

441 

134230 

43 

18 

7.72331 

286 

906296 

155 

866035 

441 

133965 

42 

19 

772503 

286 

906204 

155 

866300 

441 

133700 

41 

20 
21 

772675 
9.772847 

286 
286 

906111 
9.906018 

155 
155 

866564 
9.866829 

441 

133436 
10.133171 

40 
39 

441 

22 

773018 

286 

905925 

155 

867094 

441 

132906 

38 

23 

773190 

286 

905832 

155 

867358 

441 

132642 

37 

24 

773361 

285 

905739 

155 

867623 

441 

132377 

36 

25 

773533 

285 

905645 

155 

867887 

441 

132113 

35 

26 

773704 

285 

905552 

155 

868152 

440 

131848 

34 

27 

773875 

285 

905459 

155 

868416 

440 

131.584 

33 

28 

774046 

285 

905366 

156 

868680 

440 

131320 

32 

29 

771217 

285 

905272 

156 

868945 

440 

131055 

31 

30 
31 

774388 

284 

905179 

156 

156 

869209 
9.869473 

440 
440 

1.30791 
10.130527 

30 

29 

9.774558 

284 

9.905085 

32 

774729 

284 

904992 

156 

869737 

440 

130263 

28 

33 

774899 

284 

904898 

156 

870001 

440 

129999 

27 

34 

775070 

284 

904804 

156 

870265 

440 

129735 

26 

35 

775240 

284 

904711 

156 

870529 

440 

129471 

25 

36 

775410 

283 

904617 

156 

870793 

440 

129207 

24 

37 

775580 

283 

904523 

156 

871057 

440 

128943 

23 

38 

775750 

283 

904429 

157 

871321 

440 

128679 

22 

39 

775920 

283 

904335 

157 

871585 

440 

128415 

21 

40 
41 

776090 

283 

904241 

157 
157 

871849 
9.872112 

439 
439 

128151 
10.127888 

20 
19 

9.776259 

283 

9.904147 

42 

776429 

282 

904053 

157 

872376 

439 

127624 

18 

43 

776598 

282 

903959 

157 

872640 

439 

127360 

17 

44 

776768 

282 

903864 

157 

872903 

439 

'27097 

16 

45 

776937 

282 

903770 

.157 

873167 

439 

126833 

15 

46 

7  77106 

282 

903676 

157 

873430 

439 

126570 

14 

47 

777275 

281 

903581 

157 

873694 

439 

126306 

13 

48 

777444 

281 

903487 

157 

873957 

439 

126043 

12 

49 

777613 

281 

903392 

158 

874220 

439 

125780 

11 

50 
51 

777781 
9.777950 

281 
281 

903298 

158 
158 

874484 

439 
439- 

12.5516 
10.1252.53 

10 
9 

9.903203 

9.874747 

52 

778119 

281 

903108 

158 

875010 

439 

124990 

8 

53 

778287 

280 

903014 

1.58 

87.5273 

438 

124727 

7 

54 

778455 

280 

902919 

158 

875536 

438 

12'1464 

6 

55 

778624 

280 

902824 

158 

875800 

438 

124200 

5 

56 

778792 

280 

902729 

1.58 

876063 

438 

123937 

4 

57 

778960 

280 

902634 

1.58 

876326 

438 

123674 

3 

58 

779128 

280 

902539 

1.59 

876.589 

438 

123411 

2 

59 

779295 

279 

902444 

159 

876851 

438 

123149 

1 

60 

779463 

279 

902349 

1  1.59 

877114 

438 

122888 

_0 

J 

Ctisine 

1   Sine  [ 

Cotaiig. 

1    'J'-«-  I'M 

SINES  AND   TANGENTS.      37* 


65 


M. 

Sine 

_0_i 

Cosine  1  D. 

Tang. 

n. 

Cotang.  ' 

~-T 

9.  779463 

279 

9.902349 

159 

9-877114 

438 

10.1228S6  60 

1 

779631 

279 

902253 

159 

877377 

438 

122623 

59 

2 

779798 

279 

902158 

159 

877640 

438 

122360 

58 

3 

779966 

279 

902063 

159 

877903 

438 

122097 

57 

4 

780133 

279 

901967 

159 

878165 

438 

121835 

56 

5 

7SD.100 

278 

901872 

159 

878428 

438 

121572 

55 

6 

7804r>7 

278 

901776 

159 

878691 

438 

121309 

54 

7 

780C3A 

27S 

901681 

159 

878953 

437 

121047 

.53 

8 

780801 

278 

901585 

159 

879216 

437 

120784 

52 

9 

780968 

278 

901490 

159 

879478 

437 

120522 

61 

10 

781^34 

278 

901394 

160 

879741 

437 

120259 

50 

11 

9.781301 

-  2V7 

9.901298 

160 

9.880003 

437 

10.119997 

49 

I'Z 

781468 

277 

901202 

160 

880265 

437 

119735 

48 

13 

781634 

277 

901106 

160 

880528 

437 

119472 

47 

14 

781800 

277 

901010 

160 

880790 

437 

119210 

46 

15 

781966 

277 

900914 

100 

881052 

437 

118948 

45 

16 

782132 

277 

900818 

16-) 

881314 

437 

118686 

44 

17 

782298 

276 

900722 

160 

881576 

437 

118424 

43 

18 

782464 

276 

900626 

160 

881839 

437 

118161 

42 

19 

782630 

276 

900529 

160 

882101 

437 

117899 

41 

20 

782796 

276 

900433 

161 

882363 

436 

1 17637 

40 

21 

9.782961 

276 

9.900337 

161 

9.882625 

436 

10.117375 

39 

22 

783127 

276 

900240 

161 

882887 

436 

117113 

38 

23 

783292 

275 

900144 

161 

883148 

436 

116852 

37 

24 

783458 

275 

900047 

161 

883410 

436 

116590 

36 

25 

783623 

275 

899951 

161 

883672 

436 

116328 

35 

26 

783788 

275 

899854 

161 

883934 

436 

116066 

34 

27 

783953 

275 

899757 

161 

884196 

436 

11.5804 

33 

28 

784118 

275 

899660 

161 

884457 

436 

115.543 

32 

29 

784282 

274 

899564 

161 

884719 

436 

115281 

31 

30 
31 

784447 

274 

899467 
9.899370 

162 
162 

884980 

436 

11.5020 

30 

29 

9.784612 

274 

9.885242 

436 

10.114758 

32 

784776 

274 

899273 

162 

885503 

436 

114497 

28 

33 

784941 

274 

899176 

162 

885765 

436 

114235 

27 

34 

785105 

274 

899078 

162 

886026 

436 

11.3974 

26 

35 

785269 

273 

898981 

162 

886288 

436 

113712 

25 

36 

785433 

273 

898884 

162 

886549 

435 

113451 

24 

37 

785597 

273 

898787 

162 

886810 

435 

113190 

23 

38 

785761 

273 

898689 

162 

887072 

435 

112928 

22 

39 

785925 

273 

898592 

162 

887333 

435 

112667 

21 

40 

786089 

273 

898494 

163 

887594 

435 

112406 

20 

41 

9.786252 

272 

9.898397 

163 

9.887855 

435 

10.112145 

19 

42 

7864 16 

272 

898299 

163 

888116 

435 

111884 

18 

43 

786579 

272 

898202 

163 

888377 

435 

111623 

17 

44 

786742 

272 

898104 

163 

888639 

435 

111361 

16 

45 

786906 

272 

898006 

163 

888900 

435 

111100 

15 

4f 

787069 

272 

897908 

163 

889160 

435 

1 10840 

14 

47 

787232 

271 

897810 

163 

889421 

435 

110,579 

13 

48 

787395 

271 

897712 

163 

889682 

435 

110318 

12 

4£- 

787557 

271 

897614 

163 

889943 

435 

110057 

11 

50 

787720 

271 

897516 

163 

890204 

434 

109796 

10 

51 

19.787883 

271 

9.897418 

164 

9.890465 

434 

10.109535 

9 

52 

788045 

271 

897320 

164 

890725 

434 

109275 

8 

53 

788208 

271 

897222 

164 

890986 

434 

109014 

7 

54 

788370 

270 

897123 

164 

891247 

434 

108753 

6 

55 

788532 

270 

897025 

164 

891.')07 

434 

108493 

5 

56 

788694 

270 

896926 

164 

891768 

434 

108232 

4 

57 

788856 

270 

896828 

164 

892028 

434 

107972 

3 

58 

789018 

270 

896729 

164 

892289 

434 

107711 

3 

59 

789180 

270 

896631 

164 

892.549 

434 

107451 

1 

60 

789342 

269 

896532 

164 

892810 

434 

107190 

0 

Cosine 

,   Sine   , 

Coiang. 

j    Tang.   1   j 

62P 


56 


38°.      LOGARITHMIC 


Vl' 

Sine 

I). 

Cosine   |  D. 

Tanc. 

D. 

Cotans.  1 

u 

9.789342 

269 

9.896532 

164 

9.892810 

434 

10.107190,60 

1 

789r)04 

269 

896433 

165 

893070 

434 

1069.30 

59 

2 

789665 

269 

896335 

165 

893331 

434 

106669 

58 

3 

789827 

269 

896236 

165 

893591 

4.34 

106409 

57 

4 

789988 

269 

896137 

165 

893851 

434 

106149 

56 

5 

790149 

269 

89603S 

165 

894111 

434 

10.5889 

55 

6 

790310 

268 

895939 

165 

894371 

434 

10.5629 

54 

7 

790471 

268 

895840 

165 

894632 

433 

10.5368 

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8 

790632 

268 

895741 

105 

894892 

433 

105108 

52, 

9 

790793 

268 

895641 

165 

895152 

433 

104848] 51 

10 
11 

790954 

268 
268 

895542 

165 
166 

895412 
9.895672 

433 

104588  50 
10.104328  49 

9.791115 

9.895443 

433 

12 

791275 

267 

895343 

166 

89.5032 

433 

104068 

481 

13 

791436 

267 

895244 

166 

896192 

433 

103808 

47 

14 

791596 

267 

895145 

166 

896452 

433 

103548 

46, 

15 

791757 

267 

895045 

166 

896712 

433 

103288 

45 

16 

791917 

267 

894945 

166 

895971 

433 

103029 

44 

17 

792077 

267 

894846 

166 

897231 

433 

102769 

43 

18 

792237 

266 

894746 

166 

897491 

433 

102509 

42 

19 

792397 

266 

894646 

166 

897751 

433 

102249 

41 

20 
21 

792507 
9.792716 

266 

894546 
9.894446 

166 

167 

898010 

4^3 

101990 

40 
39 

266 

9.898270 

433 

10.101730 

22 

792876 

266 

894346 

167 

898530 

433 

101470 

38 

23 

793035 

266 

894246 

167 

898789 

433 

101211 

37 

24 

793195 

265 

894146 

167 

899049 

432 

100951 

36 

25 

793354 

265 

894046 

167 

899308 

432 

100692 

35 

26 

793514 

265 

893946 

167 

899568 

432 

100432 

34 

27 

793673 

265 

893846 

167 

899827 

432 

100173 

33 

28 

793832 

205 

893745 

167 

900086 

432 

099914 

32 

29 

793991 

265 

893645 

167 

900346 

432 

099654 

31 

30 
31 

794150 

264 

893544 
9.893444 

167 
168 

900605 
9.900864 

432 
432 

099395 

30 

29 

9.794308 

264 

10.099136 

32 

794467 

264 

893343 

168 

901124 

432 

098876 

28 

33 

794626 

264 

893243 

168 

901383 

432 

098617 

27 

34 

794V84 

264 

893142 

168 

901642 

432 

098358 

26 

35 

794942 

264 

893041 

168 

901901 

432 

093099 

25 

36 

795101 

264 

892940 

168 

902160 

432 

097840 

24 

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14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

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